Usually in upright pianos there is a spring based return mechanism that forces the damper on to the string when the key is depressed. The spring may be rusty or broken in which case only replacing it will solve the problem.
Another possible cause, the axis of the damper lever may be partially displaced, giving unsuall resistance. In that case dismounting the mechanism and reinserting the axis will solve the problem. But if it's a problem with the axis, more likeley the axis felt (or nylon) bearing (not sure if that's the rigth English word for this part) is worn out and has to be replaced (easier if it is in felt, if it's nylon it has to be a manufacturer part).
It's improbable, but there may also be something causing a partial obstruction that doesn't let the damper return to its position by force of the spring alone. If the piano is very old, dust alone can be sufficient. Carefully vacuum cleaning and very gently moving the damper back and forth with your hand it's probably worth trying. You're going to have to remove the piano's front panel to properly reach the mechanism, but's usually not to difficult.
Check also if the damper felt is in good condition. Again, it's improbable, but a piece of shreded felt may be leaning against one or two of the strings, while the other(s) remain free (if it is a 2 or 3 strings note).
BTW the screw you see is probably the one that attaches the key mechanism to the mechanism frame. If that's the case and you unloose it the mechanism is going to get free and will stop working properly and interfere with the neighbouring keys.