Perhaps your part-time job should be teaching and not showing yourself up as under par?
...
I cringe a little to think that you will be doing what I do, without even knowing jazz, probably not being able to improvise or do a song justice.
For one, teacher's should have the knowledge and skills required to both do a song justice, and improvise, regardless of whether their strong suit is classical, jazz or otherwise.
Secondly, improvisation, while a nice and valuable addition, is not required to do a 'jazz' tune justice. It might be wrong to call it jazz per se, but the music can still be very enjoyable to listen to and would be fine as restaurant background music... certainly doesnt need a "cringe"
Lastly, anyone who embarks on that path - of "restaurant" gigs, will ultimately have to learn the skills you're touting, the required repertoire is simply too large. I've played gigs like this that went for 4-5 hours, every Friday and Saturday, there's frequently the same patrons at the venue. You need to be able to improvise and arrange tunes yourself in that context or you'll soon run into problems as you'll be repeating yourself constantly.. (imagine the pile of sheet music you'd have to carry around otherwise) You must know this.. you must also know that arranging tunes from lead sheets, and in turn beginning to improvise over the chord changes is the one of the steps in acquiring these skills.
Why discourage such a pursuit by suggesting that you would "cringe" at the thought of it. And why assume that the OP would not work to gain a sufficient degree of competence there before accepting payment for performance?