Does the failure to complete the circumcision procedure perfectly, due to the doctor leaving parts of the foreskin (not exceeding 6 mm), affect the acceptance of prayers, fasting, or other deeds?

The remainder of the foreskin covering the glans should be removed after circumcision if no harm results from it, so that the entire glans is exposed.

The presence of some foreskin does not invalidate the prayer. That area must be cleansed during ablution (istinja') and ritual bathing (ghusl) because impurities accumulate beneath it.

Prayer and all acts of worship are valid for someone who has not been circumcised at all. The hadith narrated by Abu Burdah from the Prophet, peace and blessings be upon him, who said: "The uncircumcised one shall not perform Hajj until he is circumcised," is weak.