In English you have to wish FOR an object (uk native speaker here). You can wish (that) you were thinner - the "that" is optional - or wish (that) it rained less in England. But you have to wish FOR better weather or wish FOR a diet to work.
The student can WANT a computer, no "for" necessary. Almost the same meaning but want has an implication of being more acquisitive and materialistic, less wistful somehow. Often used in ab idealistic way eg wishing for world peace.
You will see "want for" but it actually has a separate meaning of "to lack". Slightly old fashioned but in reasonably common usage.