As everybody has already said, the correct sentences are "I want you to go home" and "He wants me to see you".
These constructions are called "Accusative with infinitive". Emarbe is right. They consist in a direct object (which is, at the same time, the subject of the infinitive that follows) and the infinitive together with which they are the object complement of the predicate of the sentence. It is mandatory that the subject of the first verb should be distinct from the subject of the second verb. This construction is usually used after verbs expressing:
1) wish (desire, dislike, expect, like, prefer, want, wish)
2) permission (allow, let, permit)
3) command (command, order, recommend)
4) perception (feel, hear. notice, observe, perceive, see, smell, taste, watch)
5) mental activity (assume, believe, consider, declare, find, expect, guess, imagine, intend, judge, know, mean, suppose, suspect, think, understand)
6) declaration (acknowledge, declare, deny, pronounce, state)
7) other verbs, such as: bear, help, belong to, care for, count on, depend on, help, look for, look to, make, prove, rely on, report, send for, wait for
Anyway, after the verbs "let", "make" and the verbs of perception the short infinitive is used (a). The "to infinitive" is used after these verbs only when they are in the passive voice (b).
(a) Tom let you read his book when you were on the trip with him, didn't he?
I heard her play the piano.
He made her cry for pain.
(b) You were let to read his book when you were on the trip with him.
She was heard to play the piano.
She was made to cry for pain.
After the verb "help", either short or long infinitive can be used:
My brother helped me (to) build a new house.