I am not sure what can happen if a bird strikes an engine during landing.
To narrow it down: commercial airliner (B787/A380 etc.) hitting a large bird so engine out is likely.
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I will talk from my experience as an A7E light attack pilot. It was always a frightening experience at any stage of flight. We had a turbofan engine, which is usually more forgiving to foreign object damage, or FOD, but nonetheless taking a bird of any size in the intake would almost certainly cause an engine failure. And taking a bird hit through the canopy wasn't much better. So, it can be a catastrophic event in certain aircraft, while rather benign in others.
Was in the front seat taking off on a training hop, and had wheels in the well, when we passed through a large flock of birds at the end of the runway. We were at around 500 feet AGL. It was mesmerizing. As you approach a bird that is above you it will instinctively tuck its wings and dive to get out of your way. So as we flew through the flock there were birds co-altitude with us veering left and right and then the birds above us diving close past the canopy. Miraculously we were not hit.
The other experience I have with a bird strike was when I was in the fleet rolling in on the target. I was in a tight right-hand descending turn. I was clearing the left side of the aircraft, looking over the canopy rail when I saw a speck on the canopy. I remember thinking that, "Hey the plane captains are usually better than that at cleaning the canopy," and then the speck turned into a Turkey Vulture. It impacted the fuselage a few feet behind the intake in the gun port. Luckily for me I was on a NATOPS Safety check flight and so was able to demonstrate my knowledge of a precautionary approach.
The ordinance men who fixed the gun were not happy with me. They said the repair was quite messy.
A range of things can happen.
For an airliner with a modern turbofan engine, the mostly common outcome is, absolutely nothing happens to the aircraft (although the bird's day is pretty thoroughly ruined). A sparrow can be chopped up by the fan blades & blown out the back, and the crew will never notice it. The engine won't hiccup, and until somebody sees a trace of blood and feathers on the fan blade during a walk-around, nobody will have any idea it happened.
Next case is a small bird gets ingested into the core. Chances are good the engine will keep running (remember, small bird we're talking about here), although the crew may notice a fluctuation in the engine instruments and there may be the smell of cooked (charred) bird in the air conditioning system.
In both of these cases, the engine will be inspected; the case where the bird went into the core takes longer, because maintenance will have to use a bore scope to look inside the core to inspect for damage.
The worse case than either of these is a big bird (or several) going into the engine. Depending on the specifics, the engine may chop them up & spit them out without damage, or the fan blades may be damaged, and/or the core may be damaged, and in the bad case the engine loses power, and in the worst case, it quits.
Losing one engine to a birdstrike is quite rare and is a pretty big deal, but most certainly not unrecoverable. An airliner can fly fine with one engine shut down.
If the birdstrike happens on short final, the crew might simply use more power on the operating engine and continue on to land. If it happens farther out, they might continue or they might discontinue the approach, go around, shut down the damaged engine, and then come back and land.
The really, really bad case is where you get enough bird strikes that they shut down all engines, and at that point you're a glider. If that happens at 50' over the runway, your only choice (land straight ahead) still gets you where you wanted to be anyway. If it happens farther out, well, that's where somebody gets to try to be Sully 2.0. That sort of a scenario is extremely rare – like one in tens (if not hundreds) of millions of cases rare.
You're far more likely to get hit by a semi driving home from the airport than having your day ruined by a bird going through an engine.