When sharp pokey stuff penetrates the tire and cuts a few inside layer threads/cords, but it appears okay otherwise I'll boot it with a Lezyne self sticking tube patch -- these are almost paper thin and shouldn't interfere with proper seating for most tire/tube/rim/rim tape combos.
Also, the thin patch reminds me where the tire was cut through, so next time I have a flat I'll remember to check that area to be sure it hasn't worsened. So far, so good.
And I'll fill cuts in the tread with Shoe Goo. Tough stuff, lasts a long time on my tires -- going on 7 months with my Michelin Protek Cross Max on the errand bike, which get slashed up by really nasty roads in my area filled with broken glass, metal shards and construction debris.
I'm not expecting the Shoe Goo adhesive to actually heal the cut. I just want to minimize the risk of debris infiltrating an open cut and penetrating to the tube. It works for that.
I just used Shoe Goo to fill a couple of 1/8" cuts on some 700x23 Vittoria tires that had less than 200 miles on 'em. I inspected the cuts with a 10x loupe and used tweezers to pick out a couple of tiny pebbles stuck inside. The cuts didn't penetrate or damage the thin inner puncture shield, so it should be fine. I usually give it 24 hours before riding, although the stuff dries quickly under a fan.