Several replys here have confused elasticity and what one might call stretchiness or flexibility.
An elastic interaction is one in which energy is conserved. E.g. a super ball that bounces to the same height it was dropped from.
Cork and loose strings may make a bat/tennis racket more flexible or stretch the strings for a longer duration but it does not make it more elastic. Indeed, the tighter the strings and the less cork the more elastic the interaction.