Learn Spanish (Latin America): Quiero recoger el desorden que has hecho. - I want to pick up the mess that you have made.

Hacer (to make / to do) has an irregular past participle. Instead of following the normal rules, it becomes hecho. So, "you have made" is has hecho. This is one of the most important irregulars to memorize!

Why do we need the word que in this sentence? Can I just say el desorden has hecho?

No, you must include que (that). In English, we often drop the word "that" and just say "the mess you made." In Spanish, you can never drop it. You have to literally say "the mess that you have made" (el desorden que has hecho).

Does recoger specifically mean "to pick up"?

Yes, it means "to pick up" or "to gather." In the context of a house, a room, or a mess (un desorden), it is the standard verb used for "to tidy up" or "to clean up."

Why isn't there a connecting word like a or de between quiero and recoger?

When you use querer (to want) right before another verb, the second verb simply stays in its infinitive form (ending in -ar, -er, or -ir) with no connecting words. You only need connectors for specific phrases like ir a (going to) or tener que (to have to).