Your issue is going to be either:
Local muscular endurance, meaning your body doesn't want to move at the end, movement will be clumsy legs will fel heavy, mind will be clear but nothing below will work
Cardio fatique, legs have something left but you have no air to feed them, you feel light headed and generally sluggish muscles feel strong but don't respond.
Worst case it's some of both.
Due to the fact you are able to run greater distances without effort I am going to guess it's the same weakness I have MSE.
To improve this I would advise combination of running intervals, aka sprint training or HIIT and some runn circuits.
Flat out run for 10 to 20 seconds, basically until you are ready to drop, then jog, probably slower than walking pace, for a minute or 2. the sprint will not get much longer as you should be aiming to go fast not far, recovery may be reduced as you improve, but it may be that intensity comes from higher sprint pace.
Running circuits. Basically a circuit with some running, bet this shocked you.
Find a space you can do a run, min 200m max 800m, run then do an exercise, then run again. Example is below
Each exercise 1 minute, apart from the run
Run
Burpees
Run
Press ups
Run
Sit ups
Run
Pull ups
Repeat
Rest can be added after each exercise if needed, to reduce intensity, increase time, running pace or number of circuits to increase it.
Most importantly remember not to damage your other training by doing this. It won't help you if you run brilliantly and fail everything els instead.