I periodically check and pull out the street glass shards I have been doing the same for as long as I have a bike.
Glass shards, small sharp stones, nails, drawing pins... name something I haven't pulled out of my tires!
But I never worried of patching the tire, and never had problems because of that.
My tires (even those which weren't puncture resistant) reached end of life because of wear, not because of cuts.
I just patched the tube inside, and went on.
bicycles.stackexchange.com/users/30865/L.Dutch - Reinstate Monica
The only times you need to consider the tyre is if the hole is large enough for the inner tube to poke through, or if the hole is in the thin sidewall I once had a hole that was ~1 mm across, and seemed okay.
It took about 3 flats every 200 km of riding to realise that the tube was herniating through the hole and wearing through.
A black tube and a black tyre made it hard to see the issue.
If you can't see the tube or the threads of the tyre in or through the hole, its fine.
If you can, then its possible to stick an inner tube patch on the inside of the tyre to get more wear out of it, or if the tyre's worn out anyway then its time to replace.
Your other option is to work on your technique, and try to stop riding through broken glass so much.
Not easy I know.
bicycles.stackexchange.com/users/19705/Criggie
If it's a large enough cut that the tube pokes through the tyre once inflated, then the best solution is to use a regular tube patch on the inner surface of the tyre with the vulcanising glue just like when patching a tube.
While less recommended, I've also had success just using a drop of superglue for a small cut in a tyre.
bicycles.stackexchange.com/users/40185/Carbon side up
You can use a small amount of super glue to close the area that was cut by the shard of glass.
Depending on where the cut is located you will have to re-apply the super glue every few hundred kms, but this can greatly extend the life of the tire.
bicycles.stackexchange.com/users/43660/imelhado
Some puncture kits come with a piece of rubberised canvas which is intended for glueing over a (small) tear or hole in the tyre carcass - but I wouldn't rely on it for anything bigger than a slot a few mm long - perhaps you could get away with longer on a relatively low pressure tyre but not on a racing bike.
For pinpricks or cuts to the rubber that don't penetrate into the fibre structure I've never bothered doing anything other than removing the foreign material from the tyre so it doesn't get worse.
I've never had a tyre fail because I'd not glued up or repaired damage to the tread...
general wear gets them in the end.
bicycles.stackexchange.com/users/43581/houninym
I don't bother patching cuts caused by tiny shards of glass and have never had a problem.
When touring I carry a Park Tool emergency tyre boot in case I develop a bigger cut in the tyre - but I would only use it as a temporary solution until I can get home and replace the tyre.
bicycles.stackexchange.com/users/28653/John M
There are also tire plugs which are designed to plug holes in tubeless tires that are so big the sealant does not work.
It might be overkill, but I see no reason why those would not work when using tubes.
bicycles.stackexchange.com/users/34579/Lukas
How do you deal with the small cuts in a bicycle tires main tread?  Do folks fill these small cuts?  I have continental gatorskins which I periodically check and pull out the street glass shards.  thanks!
How to patch glass cuts in a bicycle tire?
bicycles.stackexchange.com
<tire><maintenance><preventative-maintenance>
bicycles.stackexchange.com/questions/62123
bicycles.stackexchange.com/users/43628/akyeung
Cyclists in general consider that there is no need to repair a tire with a small glass cut. It is sufficient to simply remove the shard of glass. However, larger damage may require a quick fix - such as super glue, patch kit, tire plug - until you can get home and really patch the tire or replace it. As long as you remove glass and other sharp objects, smaller cuts are generally not a problem for tyres. Patch the tube inside and keep riding, and the tyre is likely to wear out rather than failing due to these cuts. Larger cuts where the inner tube is poking through are more of a problem, but can usually be fixed with a normal patch or some superglue. Some commercial products are designed to repair tyres, especially tubeless tyres, which could also be used on clincher tyres.
As long as you remove glass and other sharp objects, smaller cuts are generally not a problem for tyres. Patch the tube inside and keep riding, and the tyre is likely to wear out rather than failing due to these cuts. Larger cuts where the inner tube is poking through are more of a problem, but can usually be fixed with a normal patch or some superglue. Some commercial products are designed to repair tyres, especially tubeless tyres, which could also be used on clincher tyres.