Before this trade, the White Sox were projected to have Andrew Vaughn and Gavin Sheets both in their starting lineup, one in an outfield corner and the other at DH, with Adam Engel backing them up, then Leury Garcia (who is also their primary backup all around the infield, I think). Pollock raises the floor of their lineup considerably.
On the bullpen side, even without Kimbrel (and with Crochet out), they still have Hendriks, Graveman and Bummer who were strong performers last year; Ruiz and Burr, who were competent; plus Joe Kelly is on track to join the team later this month. And their starting rotation is pretty deep (and may provide an overflow starter for the pen).
And last summer, in addition to the Kimbrel trade, they made an affordable deal for Ryan Tepera, so they have shown some ability and willingness to upgrade the pen in-season.