If the faucet is properly attached with clips to the counter (check underneath to see if a clip is missing), then the culprit is the gauge of the sink. Adding wood is a bad idea and won't help with the issue. Replace the thin 24 gauge builder's grade sink with something more substantial like an 18 gauge, or even a 16 gauge if you can find it.

I know exactly what you're describing.....and agree with both hollysprings and brickeyee. Inasmuch as the sink's already in, would probably go with brickeyee.

However, if you have a disposal also -- which may give you the same grief in a different way -- might consider re-doing with heavier guage a-la- hollysprings.

Go with a heavier guage sink. It will cost more in the beginning but you'll be happier with the final product. I have found with my own experiences that when I try to modify something like that, it's easier and better to just go with a better product unless you can't get your sink out without a total counter top remodel.

Thanks all. For this job, angle iron it is.

This would be a tough sink replacement anyway since it was installed prior to setting the counter and is really shoe-horned in the cabinet with no clear shot to some of the clips. (And I'm not in the mood to empty cabinets, remove island counter with 45 degree entension, recaulk, etc. for a family freebie.)

I would use iron too. i would paint the iron with antirust paint.

You might find angle aluminum in your hardware store. Good enough.

I have nothing but respect for davidro1, but....do NOT use aluminum from your hardware store. It does NOT have the lasting strength/rigidity you will need.

If it's 1/8th inch thick, it's good enough. Hope this helps.

Rereading the OP, about sandwiching a wider piece of hardwood to give some rigidity, I think it might be adequate to use a few paint stirrer sticks as a trial and see if that suffices. They are about 1/8th inch thickness also. The wide nut under the faucet will spread the forces over a wider area. It seems the OP has a stainless sink with a wide flat top where the faucet hole is. The sink metal needs a little bit of extra support between the faucet's nut and the edges of the too-wide sink rim.

A flat piece f thin wood will add a minimal amount of stiffness.

You need material that is above the surface you are trying to stiffen to do much good (thus angle metal).

A steel washer to reinforce the sink would do the trick. You can buy a 1 and 3/8 in washer. Or have machine shop drill 1 and 3/8 inch hole in a 1/8" plate steel at the Machine Shop.

Steel Flat Washer, Zinc Plated Finish, ASME B18.22.1, 1-3/8" Screw Size, 1-1/2" ID, 2-3/4" OD, 0.165" Thick (Pack of 10) https://www.amazon.com/dp/B009OJHHHW/ref=cm_sw_r_apan_i_BNWCAD97RFYB3EXPBJ47

Another approach is go from the top and put one of those big flare things which I don't like. Of course you could do like others have recommended it's redo your entire kitchen

https://www.deltafaucet.com/parts/product/RP37490.html#reviews

https://www.amazon.com/Moen-113173-Mounting-Hardware-Kit/dp/B002FAY80Q/ref=sr_1_8?crid=22KHHR6KDC0ZV&keywords=Stabilization+Plate&qid=1649288758&sprefix=stabilization+plate%2Caps%2C251&sr=8-8