Which songs did The Beatles cover?

 The first two Beatles albums included around 40% of songs covered from other artists. These were mostly taken directly from their live act. 

Usually referred to as a Shiirelles' song but written by Bacharach, Dixon & (Mack) David

These covers avoided the most famous artists - no Elvis for example - and songs that had reached the higher reaches of the UK singles charts. They  were aimed at their NME-reading peers in mind. This ruled out anything 'uncool': songs by British  acts, for example, or show-tunes, for example. 

In short they were looking for hip American minor hits of recent vintage, preferably from black artists.  The Shirelle's fitted this template and provided two songs for Please Please Me.