No, lipos don't benefit from cycling like NiMh does. Do you know anyone you could borrow a diff brand lipo from and try? That would tell you if its the lipos you're using or not.

How much goes back into the lipo when you charge it? What you're experienceing is a voltage drop after those 1st few laps. If the lipos is not damaged in any way, the C rating may just not be enough for your setup. Is this the 35C rated 4S?

The other thing that would indicate exactly what's happening is to make a run with an onboard data recorder. That would give you the voltae and ampos versus run time so you would understand what the setup is drawing and what the lipo is able to supply.