I'm a newbie to this sort of thing. I've never cleaned the pressure hose or the hose barb on the fan housing, I did clean the venting last year and vacuumed everything I could see in the back of the stove and it didn't help. By the fire eye do you mean do you mean the igniter assembly? If so I've never checked to see if it was lit during this problem.
The stove needs to sense a proper exhaust flow to activate the auger and sometime a warm stove will get loose enough due to expansion of the metal to loose it and so stove will not complete that auger circuit or continue with startup. See if you can clean the hose and the adapter that feeds it off the exhaust fan. Can stick a small wire or the like into the port to clear it and if you take off both ends of the hose blow through it to clear. I would consult the manual about cleaning the eye. Can jump the circuit if needed to rule out that pest.
