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.