There are three main usages of Wish Clauses:
1. Wish + Infinitive...
When we use wish followed by an infinitive it has a meaning similar to the verbs "want" or "would like" in a very formal sense.
I wish to talk to you, please.
I wish to book a room, please.
We don't prefer a 'direct object' after wish.
wish an ice-cream
with chocolate and vanilla please.
2. Wish + Subject Pronoun + Noun...
We can use wish followed by two objects as well.
I wish you a happy marriage.
They wished me a quick recovery when I was in hospital.
I wish you a quick and easy delivery.
3. Wish + That-Clause...
When we use wish followed by a "that clause" which might be removed in informal styles, it has a meaning of regret about a situation that is impossible to change.
I wish (that) you were next to me right now.
I'm sure you all wish (that) you were rich.
We don't use wish with situations that are possible, instead we use "hope".
I hope you will write to me soon. (not I
I hope I can see her at the party. (Not I
I can see...)
Wish + That Clause Tenses
With a present or future meaning, we use past tenses.
I wish you came with me.
I wish I was there.
All my students wish they didn't have the exam.
With a past meaning, we use past perfect.
I wish I had listened to my father when he said I shouldn't drop out of the college.
I'm sure you wish you hadn't stolen that car but it's too late now.