Will God not accept prayers upon the Prophet (peace be upon him) from an heedless heart, just as He does not accept supplications?

The meaning of the Prophet's (peace be upon him) saying: "A supplication from an inattentive, heedless heart is not answered" is that Allah generally does not answer, or the response is not complete, for a supplication originating from a heart that is turned away from Allah or heedless.

As for sending blessings upon the Prophet (peace be upon him), it is accepted and never rejected, even if it is without complete presence of heart. There is great good and immense reward in it, and many hadiths have been narrated concerning its virtue, none of which stipulate the presence of heart. It is distinguished from all other supplications in that it is certainly answered.