I’m guessing an insecticide such as Malathion (but not if the plant is in bloom due to killing bees)…but I’ve never seen any insects on osmanthus fragrans, personally.
It’s a super zone 8 plant…I’ve thought of trying it in zone 6, but it probably would croak.
how many years have you been growing it? where did you get it? Is some bug damage really a problem? leaves are to be eaten to a degree it will likely recover.
1st pic: There’s a grasshopper type bug on the leaf at ~9 o’clock. The leaf cuts looks like it was done by a critter with scissor-like mandibles. You can try neem oil sprayed on the leaves as a deterrant. Or spray with soap directly on the insect to kill it.
2nd pic: The new growth coming out being a bit yellow and shriveled would concern me more. If it was just potted maybe it’s adjusting.
I bought it online in 2020 spring. It was only around 8 inches tall. I planted in a small pot and didn’t care about it enough. About two months ago, I transplanted to a bigger new pot. It looks like it has such an bug bitten experience in every spring.
Below picture was after the transplant. It is about one foot more high.