I've chased smoke from two pellet stove installs. My P68 I eventually fully resolved, and even after disassembling and moving to a new house is still smoke free, and the second one (interestingly enough, an XXV) I haven't figured out yet.
The only time you will actually see smoke is the 30 or so seconds while your stove is igniting and the whole burn chamber is full of gray smoke. That is the time you should crawl behind the stove with a flashlight and check.
The rest of the time the smoke is invisible and smells like a wood burning tool used for burning designs into a piece of wood.
Your picture has a lot of things that make me nervous.
First: It does not appear that you have a Harman specific appliance adapter to connect from the stove to the pipe, but if that thing you do have really belongs there, then I would take it off, put RTV silicon on the pipe that's bolted to the stove, then clamp that thing back onto it.
Second: there's a screw holding the cleanout cap to the bottom of the T. My first smoke chasing was due to the "professional" installers using screws that were too long, which punctured the inner liner of my venting in several places... Even if that screw didn't mess anything up inside, the screw head is probably not sealed, so I would seal that up. If you want to be able to remove that screw in the future, I recommend a high temp silicone tape, rather than the red goop.
Third: That cleanout cap itself could be the issue. I had a brand new undamaged PVP cleanout T start to leak at the cap after a while. I wrapped the connection with silicone tape. If you get the tape, it's wide enough to cover the screw and the seam with just one wrap.
Fourth: I wouldn't trust the RTV to seal that connection with the chimney liner. Like everyone else said, that doesn't look quite right. It might be hard to do with the hose clamp there, but maybe a few wraps of tape would cover that whole thing and seal it up.
I would be interested in seeing the pieces of caulk you found under the stove... Maybe it's just from the chimney itself...
Another thing that I would check is the door gasket using the dollar bill test. Everyone everywhere says that smoke can only come from the back end of the stove, but a year or so after I resolved my pipe smoke issues, I had the wood burning smell again. With guidance from folks on this website, I eventually figured out the door was not latching tight enough. I adjusted the latch (easy to do on a P68), and then the door sealed and the smoke issue went away.
When the stove is off and cold, take a dollar bill and place it between the door and stove, then shut the door on it. Then gently pull the bill out. If the gasket is sealing correctly, there should be pretty good resistance to pulling the bill out. Do this all the way around the door. This is a lot easier to do on a P68 door than an XXV door, but I was able to rule out my door on my XXV.
Lastly, check the ash pan gasket. This is my next step on my own XXV. The tadpole gasket is part number 1-00-71611 I got it from
https://pellet-stove-parts-4less.com/ for $35.00 plus shipping (actually it came from them through Amazon). I think there's not enough "fluff" left in my gasket to push back on the ash pan to get a good seal there. It's impossible to do the dollar bill test on that weird little hinged contraption, so I'm just guessing (hoping) that is my issue. I haven't replaced my gasket yet, and haven't found any instructions on how to do it, but I had a conversation with a Harman tech that remembered seeing an XXV years ago. He gave me some pointers that seem to make sense.
As I'm typing this, my wife literally said "Instead of helping other people with their stoves, why aren't you working on our stove?"
So yeah. I hope all that helps you get to where you need to go.
