Made
popular by the song of the same name as most popularized by the
band called REO Speedwagon in the 1980's era, take on the run means to run away, usually from a bad or uncomfortable situation.
It may also now refer to taking a food order to
go, aka running with it.
He suspects that you're cheating, so like a
coward instead of confronting the situation you leave town and take it on the run.
I
don't have
time to sit down and eat. I'll just take it on the run.