Clutch lever sticks

Is the throw-off on nice and solid? And the clutch engagement stop? The stop is held on by two screws, which, if they have somehow become loosened, could cause the stop to miss the stop lever for throw-off. This would allow you to run the press with the GATE RAISED, which is dangerous as all-get-out! Refer to Plate D in the Parts Book for flywheel end of throw-off mechanism, and to Plate T for the throw-off itself. The stop is shown on Plate Q, Sucker Bar and Guide.
If your Bowden Cable is intact and properly hooked-up, you should not be able to budge the throw-off with the gate raised
Go over the entire area of these mechanisms with an eye to tightening something which has somehow become loose, or loosening something which might have a foreign object in it (the Spring Sleeve, for instance).
If your press is inching while it is supposed to be stopped, that will mean you will need to adjust the clutch to compensate for wear on the spherical headed screws that work it. That is for another time.
Frank.

You were right that the Bowden cable was loose. Or, maybe the back side of the stop is worn down. Either way, I should tighten it.

Regarding the bigger problem:

The throw-off is solid. I’m pretty sure the problem is something happening inside the clutch assembly. I sent a picture of what I’m calling the clutch assembly in case I’m using the wrong term. I wish I could attach a video to show what is happening. It’s making the machine almost unusable.

Looking inside the window of the clutch assembly, there is a nut with a bold sticking out. It seems like the clicking happens when it has to push its way past whatever kind of hunk of metal that nut/bolt is on. Is that part supposed to move up and down when I engage it?

Hmm… I tried attaching the image, but nothing happened after I hit “attach selected file”. I’m guessing you can figure out which part I’m talking about. The part with the red flap.

You should download the “Heidelberg List of Spare Parts”; then you can refer to the portions, assemblies, parts, etc. by the listed names. Saves a deal of guesswork on the other end. Particularly, you will need Plates D, Q and T to help you with this
The part with the red flap is the Inverted Brake Cone. It has a Bearing Cover on the outer end; this has a green oil hole in it. This is supposed to have been oiled about every three months or so if the press is in continuous production; give it a good squirt. There aren’t supposed to be any bolts sticking out anywhere, so I’m waiting to see what kind of deal we are up against.
In the side of the Flywheel are two square windows that each have a screw and a locknut; these are the Clutch Adjusting Screws. They are replaceable if badly worn. I am telling you this so that you will understand approximately how this mechanism functions, also so that you won’t poke around and possibly further disturb something else.
First, with the motor OFF, take your 17mm wrench and see if either of the locknuts are loose. If they are, then that is at least part of the problem; tighten them for the moment while you check out the rest of the particulars.
After you have figured out how to attach your pix (I don’t know how), perhaps we be on the path to a remedy.
Frank.

https://share.icloud.com/photos/04aYyL7jk5CWX-NRtkWoIrEew

This link shows a video of what I think is causing it to catch.

I think it might get worse when the machine is cold. (We keep it in a garage with a heater, but we only run the heater while we are out there.)

Fixed!

I raised the clutch adjusting screw about a millimeter. That was it. Simple fix to a very frustrating problem.