Running torque is also called the "friction torque". It's the torque required to make the nut or bolt turn while the nut is "loose" you might say. With a locking nut, the locking feature (normally either a plastic insert, or a distorted bore section of the threads that directly squeezes together on them) takes a certain amount of effort to overcome.
Since torque value is an attempt to measure the stretch in a bolt as the rotary sliding wedge of the threads is applied to pull the two ends apart, any rotational resistance forces that are added, not directly applying the clamping force to the part, will skew the final result.
If you torque to 100 inlbs, and it takes 30 inlbs of torque just to make the nut turn because it's a lock nut, when the wrench trips at 100 inlb you will only have actually applied 70 inlbs worth of "stretch" to the bolt. So if you measure 30 inlb while spinning the nut down before it clamps down, you add that to the 100 inlb value, and run the wrench to 130 and will have the proper 100 inlb application to the bolt or stud.
The proper procedure is to run the nut or bolt down to maybe half a turn to contact (washer still loose). Measure the torque required just to move the nut the last bit before the nut makes contact, and add that to the total. If the nut spins freely because it's a castellated nut and has no friction device and you can spin it down all the way with your fingers, don't bother because the value will be too low to measure.
This is the problem with preset click-over torque wrenches that can only measure a limit that is set beforehand, when torquing friction-locking nuts. When a procedure calls for that running torque part, it's better to use a dial indicating or other continuous indication torque wrench that lets you do the running torque observation and addition to the final value on the fly.
When a torque is for a critical part where precision is essential, it's also important to get to the final torque in one motion value without stopping. That's because if you stop mid way just below the final value, the torque required to get started again will be high, and may be higher than the final value, and you may get a false "I'm there" indication when you really weren't there yet. Better to back off half a turn and do it again.
If a torque spec says to lubricate threads, this is critical as well, as the friction of the thread and nut interface itself, while the clamping force is being applied, can skew the final result. On aircraft cylinders, where a precise torque application is vital, it's standard procedure to use lots of the recommended oil on the nut and threads to get the thread friction, and the nut to flange friction, as low as possible. Piston engine cylinder nuts generally don't use any locking devices (or washers) at all, and totally depend on clamping force to avoid backing off, and if done properly, they never do.
In the procedure you're quoting, the objective is to seat the "plug" for some reason with a larger than final force, then back off and re-tighten to a pretty low value of 15-20 (this sounds like a procedure intended for sealing purposes). Factoring in the running torque will be critical to achieving the 15-20 inlb final value.