Is the prayer of one who makes mistakes, such as not saying the Takbir after bowing, accepted?

Perhaps the questioner means the takbir (pronouncement of "Allahu Akbar") upon bowing, not after it, because there is no takbir after bowing. Rather, the worshiper says: "Allah hears those who praise Him" if he is not a ma'mum (follower in congregational prayer). The takbir for bowing and all takbirat al-intiqal (transition takbirs) are among the obligatory acts of prayer according to the most preponderant opinion, so it is not permissible to intentionally omit them. Whoever omits them forgetfully should perform the prostration of forgetfulness (sujud as-sahw), and his prayer is valid. The matter of acceptance is with Allah.