It would be like farting at a hurricane and expecting it to shift trajectory.
If you fart on asteroid far away enough from the Earth, tiny change of it's trajectory would be enough for it to miss the Earth.
Yes, of course, it's a trivial effort, left as an exercise for the reader.
The issue with mining any kind of resources from another planet is the immense cost of returning them to Earth.
You'll find that 100% of Space Nutters will always look at you like a Special Ed student when you mention "returning to Earth". Obviously, Space Minerals are to be used by Space Humans. You know, those vital populations of humans in free fall exposed to healthy Space Radiation and osteoporosis, huge demand for Space Diamonds there.
It's religious circular reasoning couched in pseudo-scientific language so that the engineer part of the brain doesn't notice right away.
So you are one of those "how dare we insignificant ants think we can change the course of mighty nature or avoid that asteroid that God sent here to punish us". Give me a break. Of course it would be possible to nudge a rock a bit if it looks like it will hit earth. And it would be worth all the money in the world if we can do it. Depends on the precise circumstances of course, we might not be able to prevent all possible asteroid impact scenarios, but we might just be able to divert the majority of dangerous asteroids... if we prepare for it.
No, otherwise washing your hands before surgery would be changing the course of nature too, wouldn't it?
You don't seem to understand that all the nonsense you're talking about is simply not possible, not now, not ever.
It's space garbage for children. Sometimes I read the stuff I wrote 25 years ago and it's the same shit you're writing now, word for word. Like a religious homily, or sermon.