Yes, this is the tensioner, you can probably straighten it enough for it to work, just make sure both rails are flat and straight from the eyelets to the 90 degree angle of the rubber coated part. Be careful to install the bolt and nut so the tensioner doesn't move, but that the derailleur pulley still turns freely. Without the chain retainer the chain may slip off the chain ring when folding or unfolding the bike. The retainer bolts on the the round boss on the top of the chain ring.

A little play in the gears is okay, but you can replace the components easily enough. The original parts are Shimano Sora, which are inexpensive, but you can install pretty much anything you like.