Collectively, Indians left-handed relievers have thrown 42.2 innings and allowed 39 earned runs good for an 8.23 ERA and have a 1.69 WHIP, 5.3 BB/9, and 10.8 K/9.
The numbers don’t lie. While they are racking up strikeouts as a group, they're giving up way too many free passes. Hitters are making solid contact against them much too often.
Rich Hill has been the primary lefty the Indians have used out of the pen, but he has been largely ineffective for several weeks now. 

He hasn't had much success against lefties as they are hitting a solid .258 with a good .820 OPS off of him. He hasn't been very reliable and has to be on borrowed time with the Indians unless he has a miraculous turnaround.
Nick Hagadone has received most of the remainder of the left-handed relief opportunities and his performance has been spotty.  There is no doubt about his stuff as he displays a mid-90s fastball and very good slider, but his main problem has been command and consistency.  He has actually held lefties to a .230 average and .670 OPS, which are good numbers, but the walks that have really hurt him.
Scott Barnes and David Huff have also had limited opportunities and both are no longer in Cleveland. 

Barnes has since been sent to Triple-A Columbus and Huff is now pitching in Triple-A for the Yankees. Even though their sample size was small, both showed very little in the chances they were given.
Giovanni Soto, who has been pitching well at Triple-A Columbus, might have been called up by now if it weren't for a lower back injury he's been battling all season.

Columbus lefty starter T.J. House might be another option if the Indians become desperate enough, though it appears unlikely they would use him in such a role.
Beyond those two, the Indians really don’t have another lefty relief option in the minors. The trio of Hill, Hagadone and Barnes are going to have to turn it around quickly, or they'll have to make a trade.
If the Indians want to have sustained success this season they are going to need to be able to matchup late in games to get tough left-handed hitters out in key situations. 

Some of their right-handers like Allen and Shaw perform well against lefties, but any manager would still prefer a matchup lefty to make the hitter uncomfortable.
It is a position the Indians will likely have to address at some point over the next several weeks via the trade route. If and when they do find that suitable left-handed relief option they can rely on, the overall performance and effectiveness of the bullpen could take off.