do you have something like cut one's arm or leg?
We have a saying, if you want to call it that, to the effect that you should "cut your losses." It's kinda like "don't throw good money after bad." They both mean you shouldn't hesitate to take a certain (but smaller) loss rather than forgo doing so and try to hold out in hopes of a greater gain (or avoiding any loss at all).
They don't quite fit your example, but that's the closest I can think of.
Personally, I don't subscribe to such advice. I kinda like to be "all in." Otherwise you would just cut and run from a fight after you took one punch.
Speaking of which "cut and run" is another relative here. When English ships would see pirates approaching their anchored ship, the would often "cut and run." That is, they would just cut the rope to their anchor (thereby losing the anchor forever) so that they could "run" as fast as possible.