For example if a student wishes for a storm because he wants his classes to be suspended. His main intention is to have no school but not necessarily to have a storm.
Not sure if this makes any sense. I'm pretty sure there's no word for it but I don't know because English is not my first language.
if a student wishes for a storm because he wants his classes to be suspended, the class suspension is the main intention and the breaking storm is simply a godsend.
godsend: a desired or needed thing or event that comes unexpectedly.
Your motivation for wanting the storm is to avoid school. If you rented a small plane and the agency asked you why you wanted it, you could honestly respond "I'm going to seed the clouds so I can make it rain", but your ulterior motive would be to get school cancelled.
The distinction between "motive" and "ulterior motive" is that that the latter -- ulterior -- means "hidden", and has the connotation that you don't want anyone to know about it; and that if someone asked you'd why you were doing what you were doing, then you'd only present the surface, or ostensible, reason for your action.
A non-ulterior motive is usually not hidden, and whether you tell people about it typically hinges on whether that information is relevant. For example, if you walked out of your house, and your little brother asked you "where are you going?", you might respond "none of your business!". In this case, your motivation isn't "hidden" -- for example, you'd be happy to tell your buddies where you're going -- it's just not relevant to your little brother.
Another way to identify the distinction is that someone with an ulterior motive will almost always have a plausible surface motive which would justify their actions if anyone asked (and may have some actual, if secondary, benefit to them), but doesn't reflect their true objectives.
Inadvertent: not resulting from or achieved through deliberate planning. "an inadvertent administrative error occurred that resulted in an overpayment"
I would label it as collateral damage.
relating to something but less important than it
