with trigger shifters, the tiny screw to take off the cover plate can sometimes be underneath, but once off, it should expose some of the innards and allow better spraying of stuff inside to clear out the gummed up grease. You may or may not be able to see the mechanism with the cover off.

I suggest looking up a diagram of your shifter model, it will make it easier to see how to remove cover.

oh, and you may need a tiny set of screwdrivers as the heads are often small with shifter cover screws.