The lever near the faucet that controls the push pull drain actually lifts and raises a mechanism inside the drain that pushes the drain stopper up and down. If this mechanism becomes clogged it will not be able to reach the bottom of the drain stopper to push it up.
Repair the clogs by first pulling up the drain stopper with your fingers. This can be difficult if the drain has been stuck for a while. Try to rotate the stopper if possible to loosen it. Wedge a screwdriver in between the stopper and the sink and try to pry it up. Be careful not to scratch your sink. If this doesn’t work, press a high performance suction cup over the stopper and pull it up.
Once you’ve pulled out the stopper you’ll need to completely clean the drain to remove the hair, soap scum and hard water buildup that caused the clog in the first place. Do this by pouring 1/2 cup baking soda down the drain followed by 1 cup white vinegar. Let it sit and bubble for 15 minutes to give the solution time to completely loosen the debris then flush the entire drain with one gallon boiling water.
The clevis is the metal bar that extends from the lever and has a joint called the pivot rod that connects it to the mechanism in the drain that pushes the stopper up and then lowers it down again. If the clevis or the pivot rod isn’t installed properly or it is becomes dislodged, it may cause the stopper to get stuck in the down position.
In order to fix the clevis and the pivot rod you’ll need to access them under your sink. Unscrew the nut that holds the pivot rod into the drain and remove the rod. Clean it off. Drop the stopper back into the drain so that its hole faces in the direction of the pivot rod. The rod will go through the hole to hold the stopper in place. Replace the pivot rod and tighten the nut.