If a plane has been delayed at the origin how does it make up for the lost time? Does the pilot need any special permission from ATC to increase the speed so that it can cover the lost time or has the pilot the power to make such kind of decisions?
According to ICAO Annex II (Rules of the air) 220.127.116.11(b):
Variation in true airspeed: if the average true airspeed at cruising level between reporting points varies or is expected to vary by plus or minus 5 per cent of the true airspeed, from that given in the flight plan, the appropriate air traffic services unit shall be so informed.
So the pilot in command has the right to increase speed, just as well as the air traffic controller has the right to limit it. However, as airliners normally cruise fairly close to their maximum speeds, this isn’t very effective. Recovering lost time by increasing speed is at best 1 to 3 minutes per hour.
Better ways are to request possible shortcuts from ATC and change altitude for better wind and temperature.
That's a common scenario, that you're pushing off the gate late and would like to make up time. There are a number of ways that making up time can happen, although how much of that the pilot can affect doesn't often amount to a huge impact.
The flight is scheduled for whatever length of time, gate-to-gate. This accounts for the sorts of delays that the planners foresee when they're building the schedule: it's winter, so the aircraft may need to be de-iced. It's a busy time of day, so they plan on a slower taxi than simply driving out to the runway & taking off. The flight is usually scheduled around a moderately worst-case headwind/(lack of) tailwind. And so for the 1+00 (in the air) flight, you might be scheduled for 1+30 gate-to-gate.
And if some of the negative factors don't materialize, then you'll under-fly the schedule, which is nice. Maybe the delay at the gate caused you to miss the "morning rush hour" and you did get to taxi right out, no line or delay, and take off. There's no precipitation, so no delay for de-icing. The tailwinds are better than average today, so instead of 1+00, your flight will only be 0+55 in the air. All good things!
But none of that is really under the pilot's control.
The two places that the pilot can most affect things are, going faster, and getting shortcuts. Going faster essentially means using more gas, and sometimes guys are willing to do that... the high-school tour group is in danger of missing their connecting flight & being stranded in the hub for the night, let's go fast & get them there! Another version of going fast is flying at a more "speed efficient" altitude, usually around FL290 or FL300. If winds are all the same, you'll generally get more speed there because your maximum indicated airspeed (Vmo) and limiting Mach number (Mmo) are very close around that altitude. The tradeoff is, however, that you are burning more gas by doing this compared to going higher.
If you have the gas to burn, this can typically work.
Also, you can request shortcuts. Depending on the filed route and what's going on in the airspace, this may get significant savings. From the West Coast to DFW or IAH, you're typically routed around the north or south side of White Sands. But, if it's Saturday night and the restricted area is cold (not in use), then you may be cleared direct right through it & save some significant time. Of course, requesting shortcuts also saves gas (unless it takes you off a wind-optimized route -- rare but possible), so there's no real reason not to do it on every leg.
ATC may or may not cooperate with your plans. If the restricted area is hot, then you don't get the shortcut through it. If you're inbound to a busy terminal area (DFW, LAX, IAH, BOS, etc) then ATC probably has a sequence in mind, and they won't approve anything that changes the orderly flow of aircraft into the arrival. In fact, they may assign you a speed in order to fit the sequence together.
Other than that sort of planning, though, they typically don't mind if you fly fast, and if they aren't too busy, ATC usually approves requests for shortcuts that don't have you driving through restricted areas (or other special use airspace).
How much time can you make up as a pilot? If you really want to burn the gas, you might make up 5-10 minutes by going fast on a typical 1-2 hour flight, but those extra minutes are getting pretty expensive when you do that. Shortcuts may give you another 5 minutes or so, in the best case. (Longer flights tend to be more direct, as a % of the time, so you can't get the bigger & bigger time savings on the longer flights with shortcuts as you do with speed, compared to the short flights.)
But when a pilot tells you that he's going to make up 20 minutes off the planned flying time, most of that will probably be already baked in (tailwinds, no delays, etc) rather than things that he's doing differently than he otherwise would. (It's still a nice line on the PA, though.)
And some days, none of the above works the way you hope, and the late flight doesn't get any earlier, or even gets later, despite your best efforts. Speed, time, and distance can just be a bear, sometimes!
"Block" times are most often padded to cover any delays like this. FlightStats has a good explanation of what goes into calculating block times. The short version is that carriers publish the "quoted" time it takes to go from block-to-block. Pushing back from the departure gate to pulling into the destination gate. As shown in the FlightStats link, some quote a shorter duration for the same segment.
On the destination side, nobody ever complains that their flight landed early. Problems for the airlines that do this is that when a plane is on the ground, you're paying money to the port instead of making money flying the plane.
Note that not all airlines have the same trip duration listed for a given flight segment. This can be tough to determine because such overlap generally requires one end be a large hub, and on the East coast US, these large airports are often slot controlled.
Padding block times helps accommodate WX (Weather), headwinds, tailwinds, etc. All at the expense of $$$.
Interesting info on actual performance vs. Airlines published block times: http://projects.fivethirtyeight.com/flights/
Also background on how that infographic is built and what factors into the decisions: http://fivethirtyeight.com/features/how-we-found-the-fastest-flights/
On NMOC (Network Manager Operations Centre) there is one member of each airline, and in case of fleet delays, due to weather for example, they discuss and instruct the ATC to expedite one or some flights, clearing them to fly direct, and sometimes, overtake leading traffics on the queue.
I am not a ATP, but I would be surprised if commercial flights went faster to make up time, even if they were delayed. The reason for this is that it is much more expensive because you burn fuel faster. To go at top speeds it is not a linear relationship and you could spend a lot of money to get a marginally faster time of arrival.
One free way to increase speed is to take advantage of winds at altitude. So, for example, if there are known to be favorable winds at 40,000 feet, you increase altitude to that level.
You do not need "permission" to go faster. The pilots are the bosses and ATC are the servants, not the other way around. If a pilot wants to go faster, he goes faster, and notifies ATC of the change and they adjust what they have to.
On some planes (airplus, sorry no reference) there is a “sprint mode” that uses more fuel but goes faster. Airlines may therefore decide to time their flights not using the “sprint mode”, so as to save fuel costs, but be willing to use it if they are running late.
Flights are also timed so that can keep to timings even if the wind is in the “wrong” direction, therefore often a flight can go faster than the planners assumed.