For core strength to have any effect, your saddle must be far enough back for your bum and thighs to counterbalance your cantilevered torso over the forward (loaded) pedal.
It’s like a see-saw. If the distance between the see-saw’s pivot (forward pedal) and one child (saddle) is too short, no amount of plank stiffness (core strength) will prevent the other child (your hands) going down hard.
You can fool around with gloves and thick bar tape and handlebar height and all the rest, but the definitive way to fix this is to get the weight off your hands. The only way to do that without ruining your aerodynamics with an upright position is to get the saddle back.
Adequate core strength will follow automatically in a matter of weeks.
See “SADDLE SETBACK” on page 10 of
this fit guide (PDF) by the man who designed Spa Cycles frames (the gent in the photos).