Questions & Answers about Мы хотим сыграть в футбол.
This is an aspect question.
- играть = imperfective
- сыграть = perfective
In this sentence, сыграть suggests playing one game/session from start to finish. It sounds like:
- we want to go play a game of soccer
- we want to have a soccer match
If you said Мы хотим играть в футбол, it would sound more like:
- we want to play soccer in general
- we want to be playing soccer
- we want to engage in the activity, without focusing on one completed game
So сыграть is natural when the idea is one occasion of playing.
The prefix helps form the perfective verb.
With играть, the action is seen as ongoing, repeated, or unbounded.
With сыграть, the action is seen as a whole event. In this context, it often means:
- to play a game
- to play one round/match
- to complete the act of playing
So с- here is not really something you translate separately as a word like with or from. It is part of the verb form that changes the aspect.