Is it permissible for me to benefit from a recommendation given to me to obtain a job, given my fear that not being accepted for it will lead to my failure?

If you are qualified for the position and succeeded in the promotion exam, there is no blame upon you for the promotion and its benefits, even if someone intervened with favoritism without your request, as long as this did not affect your entitlement to the promotion. You should abandon these doubts and strive in your work.