Is it safe to ride with a hole in your tyre?

Maybe as a spare, but I wouldn't ride it long distance or certainly not in a race situation. Put a piece of old tube to cover the hole from the inside to avoid the tube easing out of the gap. Never gonna be good as new, but you can get some more training miles out of it.

As a fellow cheapskate, I'd say you might ride it on daily training rides, but keep a very close eye on the damage before every ride. If you see any signs of the cut enlarging, or the tube showing through, replace it before you have a blowout.

You can just replace the rear tire if you wish; I've run mismatched sets often. This makes a statement that you are a tightwad and proud of it.

Road hazard damage isn't covered by warranty. In facts, cuts in road tires are fairly common. I had a Michelin Pro Race ($50 at the LBS) cut through on my second ride a couple of years ago. That's when I decided to try a different brand.

Not sure how strong your OEM tires are, but there likely replacements that offer more puncture resistance than the Rubino Techs. Conti GP 3000s have been virtually "cut-proof" for me. They tout "5 plies" and 430 threads/inch casing; there are other choices with strong plies and casings as well.

Also, it's been my experience that lowering pressure from max sidewall rating to 100 psi or less reduces problems with cuts and punctures regardless of tire brand.

"a few mm" sounds a bit big, but I patch my tires from the inside all the time.

If the hole is small enough I'll just use a tube patch, but if it's bit bigger I'll tape a small piece of thin plastic over the hole on the inside.

I often patch very small side-wall scuffs, just as a precaution.

I've never had a failure doing this.

My test is that, if I can feel a bulge when the tire is inflated, some of the cords in the caseing have been cut and I replace the tire.

It usually doesn't take much of a cut for a tiny bubble of inner tube to extrude through the tire. I stick a piece of duck tape on the inside to keep that from happening.

i usually inspect my tyres (conti gp 4000's) every week. the best way i've found is to deflate the tyre and work your way around taking out embedded glass with a fine pick. any major holes, eg 2mm or more, i glue up with superglue and i havent had any problems yet.