Is what I pledged to Allah—to abstain from masturbation—considered a vow that necessitates expiation and punishment, given that I committed the act weeks after making the pledge?

Whoever makes a covenant with Allah to abandon a forbidden act must fulfill his covenant. If he breaks the covenant by doing what he covenanted with Allah to abandon, he must repent to Allah and offer an expiation for breaking an oath (kaffarat yamin). This involves feeding ten poor people, or clothing them, or freeing a slave. If he cannot find any of these, then he must fast for three days. It is also recommended to stop the practice of masturbation.