Is it permissible to benefit from a medical specialization certificate after obtaining it by paying a sum of money to an official, with the aim of securing the specialization and avoiding compulsory military service that might hinder performing prayers, knowing that such payment is common in the country and may have led to usurping the right of another person?

If the conditions apply to you and you cannot join the specialization except by paying money to the official, then there is no blame upon you for paying, and the sin is upon the official, because you are paying to obtain your right. However, if the conditions do not apply, then you have committed a forbidden act, but this does not invalidate the certificate because you have successfully completed the training, and your issue is less severe than cheating in exams, so there is no harm in your benefiting from this certificate.