How many miles does the tire have on it? If the tire has been used enough you may want to just replace them. You can upgrade and get a set of Michelin Commander II's or you can do a WTB post for someone's original Kenda's. A lot of people(myself included) replaced the stock tires pretty early on so they'll be sitting around and can be had on the cheap. You can also pull the screw out and plug the hole. It's very easy and a quick search on youtube will give you the information and confidence to get it done.
The dealer found a nail in the center of the rear tire at the 500 mile service and it wasn't noticeably leaking. The dealer will not patch the tire so I rode it home, plugged it and put some Tire Slime in. So far so good.
Been there done that.... I think I had 200 miles on mine.... Plugs are for limping home from a trip.... New tires are for screw holes !!!! Expensive, but your safety is well worth it....
Patch it AND that is all you will think about every time you go into a corner aggressively....
Easy formula, front has any damage, replace as soon as possible. A plug is to get home. On the rear tire, it depends on mileage, where the screw punctured, and if it can be patched. A plug is not a good idea on a bike, car is okay. Slime, Fix a Flat, etc, not good and should be used in a rare emergency. If you are half way or more through the tire's expected life, replace. In the long run it is safer.
I patch mc tires all the time,unless it has slime or fix a flat,then I won't touch it,it's a mess.
Patch with a patch and plug combo,and put some ride on in your tire.
Slime and fixaflat is latex based,ride on is water soluble.
I have it in my tires due to the fact you can remove the nail and it will seal better than slime.
Take your wheel in and pay cash,that will removed the liability on the tire shop and they'll probably do it.
Check out ride on tests on youtube,this guy drills 1/4 holes in a mc tire,and no air loss.
My first modern bike, after years on my grandfather's 1980 CB650C, was a Yamaha YZF600R. Hadn't had it a month when I went out in the morning, and the rear was flat. Low-grade cussing. Patched it and rode that tire for thousands of miles. If you've never patched a tire before, that's different, but if you have any experience, patch the tire and see if it feels solid. If it does, my experience shows it can work well.
This is one of those topics that gets discussed over and over again, with some saying "never ride a plugged tire any longer than absolutely necessary." And I get it - tire performance is critical. Heck, just a little bit ago I was telling someone not to risk riding on a tire showing weird wear. But in my experience, the plugged tire showed no other signs of a problem. If you plug and it starts to wear in an unusual way, or heat up more than normal, or anything like that, change the tire. Monitor your tire extra carefully to be sure you catch it if these conditions develop. But if they don't, repaired tires can work fine. Once I gave the plugged tire a couple of hundred miles to prove itself, I rode it hard.
I've plugged lots of tires with the string type plugs.Never a problem. You stick 'em way in ,twist 1 rotation,pullout just enough to cut the tool loose. You now have 4 strings in the tire. If it's almost wore out ,I'd just buy a new one though.
It may not be right through, screw it out and see.
Your tyre looks almost new. If it's punctured two things need to happen, ingress of water to the cords needs to be prevented and the leak needs to be sealed for life. A plug is the common way to seal provided it is done properly and an internal patch is the solution inside the tyre to ensure no air leak in the future. However, if you screw that screw out and the hole is barely perceptible you can do an excellent job by forcing a clean pointy pliers into the hole and spreading it open then flush it with superglue right down to the cords and quickly remove the pliers. Put pressure on the hole while the glue cures, it won't take long on rubber, just 10 seconds or so.
Superglue is superb adhesive for rubber. Still patch the inside though as belt and braces.