Should you ever keep a kicker with a pair?
No — never in Jacks or Better. Keeping an ace kicker with a low pair drops the exact EV from 0.8237 to 0.6753 per coin: a 0.148 loss on every single hand, one of the most expensive habits in the game.
The exact numbers — 3♣ 3♦ A♥ 7♠ 9♦
EVs are exact — every possible draw enumerated, not simulated — at max bet on 9/6 Jacks or Better. Check it yourself: the analyzer runs the same solver in your browser.
See all 32 holds of this hand ranked →Why
The kicker occupies a draw slot: with three draws you have far more ways to hit trips, two pair, or a full house than with two. The ace 'insurance' blocks about a third of your improvement chances while adding only a weak pair-of-aces possibility.
No chart line ever holds pair + kicker — the pattern doesn't exist on any correct strategy list. See the full solver-optimized strategy chart, or drill hands with exact-EV feedback until plays like this are automatic.