That rod is called a push rod, at the very bottom of it through that little hole
where you can't see is the camshaft where the other end of that there push rod
rides on certain egg-shaped lobes that make it go up-and-down as the engine turns.
This push rod pushes UP on a lifter, the thing that it keeps popping out from under is called a lifter.
On the other end of that lifter is your valve.
The first thing I would do pretty much regardless is after you figure out how to adjust it,
is replace that push rod because if it even got the slightest bit of damage and you put it all back together
it could end up tearing things up, probably not but they're not that expensive and better safe than sorry.
Beyond that I have to leave it to the experts.