A minute later a ricochet from a Logan Crawford shot again fell to Kenny before a long range Sam O'Regan effort went over the bar.
But perhaps the best chance of the half fell to Andrew Cooper. A pull back from a strong Kenny run was skewed by Cooper, missing the target.
"It's frustrating," said Lowry. "whenever you have chances to score (and don't). We could have had three in the first half. Andrew Cooper's, I think the keeper's done ever so well with the first attempt and the second one's fallen to him, and I don't know if he's rushed it or scuffed it or maybe got a bad bounce.
"Maybe he just rushed it, but we rushed everything today - we didn't play football - it was like schoolboy football today."
The second half was only five minutes old when the visitors scored the game's only goal, a long corner knocked back into the six yard box for Jared Colligan to rise unmarked and beat City keeper Tom Pamment.
"Again, defensively, we do well, in midfield we do well, but it's all about composure."
Tauranga started with not-100-per-cent-fit Jerahl Hughes on the bench alongside Dean Styles, returning from suspension, Josh Nelson, who sat out Saturday's draw with Glenfield Rovers, Harry Gawtry and Jordan Culpepper.
After a strong performance on Saturday, Logan Crawford was rewarded with another start.
Lowry has no problems with the strength of his squad but said finding the right combinations it the key.
While the cup loss was disappointing, it does at least remove a distraction to the club's main goal.
"Survival in the premiership is more important to the club than the cup, but saying that, it's always nice to have a good cup run - we had a good cup run last year and it would have been nice to have done it again."
Next Sunday it's back to Northern Premiership action when Onehunga Sports are the visitors to Links Ave.
