Jeff gave his definition of an awkward connector hole. But, I don't recall reading one from Tom. Tom, what is your definition of an awkward connector hole?
Awkward to me implies a sort of forced strategy or limited options. Also, it may be a hole with an indiscernible strategy - not one that is hard to figure out but one that is not capable of being figured out. Like Jeff, it might also mean ground that is overly penal (i.e. too narrow or steep) with little chance for recovery. It likely means a hole that is crammed in. However, several posts on here seem to be equating awkward with dull or boring , for example holes on dead flat ground. The first and 18th at TOC were mentioned. In my opinion, those holes aren't awkward. They may be more boring than the rest of the course though.
I don't know if they are connector holes, but here are a couple of holes that I find awkward.
#13 Tobacco Road - A 90-degree dog leg par 5 with a Dell green? Really? The first two shots are totally meaningless. It is a layup off the tee, a layup to 100-130 yards, and then a wedge to a dell green. Not to mention it is the third 90 degree dogleg par 5 of the course. It would be less awkward if a risky drive got you to 160-180 yards so that you had a chance to go for the green. Perhaps a shorter hole that was a par 4. But, perhaps the course had to be a par 72.
#9 Kingsley - The more I think about this hole the less I like it. It really seems to be shoe horned into a limited space. It would almost be better to just play 17 holes. It plays as an afterthought even though it wasn't. Perhaps too much thought was put into trying to make it work - hence the two different angles from the two teeing grounds. Getting from the lower ground of 8 green back up to the clubhouse seemed to pose a problem.