Is it permissible to withhold an agreed-upon sum from a carpenter—due to his manipulation of wood quantities and poor craftsmanship—in order to use it to repair the damages resulting from his work?

The situation is one of two possibilities:

First: If the worker has completed his work according to what was agreed upon, then you must pay the remaining amount, and it is not permissible to withhold his right as long as the work conforms to the specifications.

Second: If the worker has violated the terms of the agreement, then he must complete what is missing and rectify the defect. If he does not do so, then you have the right to deduct the value of that from his due, based on the hadith: "Muslims are bound by their conditions."

The contract must be considered. If the worker has violated it, then you have the right to compel him to complete it or to deduct the value of the deficiency. However, this must be proven by evidence, as a mere claim is not sufficient. The default is that you pay him what remains of his right unless his breach of contract is proven.