A decent diagram here.
https://americansocietyofarmscollec...7-Variations-of-the-Model-1885-Winchester.pdf
Are you comfortable pulling the action apart to inspect the mechanism of the trigger? Cleaning and re-oiling as required? It may be as simple as a bit of dirt or dried oil, keeping it from working.
The basic operating principle, is that you push the trigger forward with your thumb, to preload the fly/hammer (may not be using the correct name) against the secondary spring in the trigger mechanism. The 'set sear' engages the fly/hammer at full travel (more or less) forward. When the trigger is pulled, it releases the fly/hammer against the sear. The screw varies the engagement of the 'set sear' on the fly/hammer.
If it bothers you and you become frustrated with it, PM me, and I will send you a mailing address where you can be sure of it being safely disposed of!
In all seriousness, it is most liable to be a simple problem of needing a decent clean and lube, though, in more or less a century, it could have suffered unkind treatment along the way, or, simply, worn, and need some attention.
Someone else may be able to chime in as to whether the set mechanism will operate as designed, if the hammer is not at full #### first.