Is it permissible to delay the prayer slightly after its time has begun in order to sleep on one of the sides for the purpose of conceiving, provided that it is performed before the time of the next prayer enters?

You bear no sin for delaying the prayer after the Adhan, as long as you pray it before its time expires. Prayer is permissible throughout its entire time, and it is not obligatory to perform it at the very beginning of its time. This is from Allah's mercy upon His servants. Praying at the beginning of the time is better, except for the 'Isha prayer, for which praying at the end of its time is better.