how to tighten the nuts inside the U shape under the countertop in the kitchen
That type of faucet typically comes with a special deep socket tool to make fastening easy. Without that tool (and they are not "universal"; each faucet make/model may have different size/style) you are left to making some other tool work as best as you can (long handled plier, basin wrench, crowfoot wrench, adjustable wrench, etc.)

I would never attempt to tighten that up with those hoses in the way, remove them first to facilitate access.

You say that the mounting surface is "very soft and difficult to tighten the nuts". I am afraid that will be a problem for you. You really should remove the entire faucet and repair/reinforce the damaged sink.