I suspect that the APUs contribution to take off perf (if any), is through its use to supply air to the ACMs (cabin air). This on some types might allow a "bleeds off" take off, providing of course anti-ice is not needed.
Taking bleed air from the engines reduces thrust through the following effects, as I understand it.
1-Thrust is determined by M(V2-V1), where M is the mass of air, V1 is the inlet velocity and V2 is the exit velocity (this is simplified for brevity)
Bleeding air out of the compressor reduces the Mass of air going through the whole engine, reducing thrust.
2*-It reduces the compressor air available to cool the turbine blades, taking the engine closer to temp limits.
So if you get that bleed air from the APU, your engines could conceivably run to a higher output.
There is no shortage of people here more qualified than I, and I would be happy to be corrected if this is wrong.
*Edit: I forgot that it is assumed the mass of fuel added cancels out the mass of the air bleed, so, I guess its more of a temp limitation.