Learn Italian: Puoi avere l'anticipo, a patto che tu concluda il progetto. - You can have the advance, on the condition that you conclude the project.

The phrase a patto che (on the condition that / provided that) introduces a condition that must be met. In Italian, this type of conditional conjunction always requires the following verb to be in the subjunctive mood. Therefore, the normal present tense concludi becomes the present subjunctive concluda.

Could I use purché instead of a patto che in this sentence?

Absolutely! Both a patto che and purché express a condition meaning "provided that" or "as long as", and both require the subjunctive. Saying purché tu concluda il progetto would mean exactly the same thing.

Why is the pronoun tu included? Doesn't Italian usually leave subject pronouns out?

Usually, yes. However, in the present subjunctive, the verb forms for io, tu, and lui/lei are often identical. For the verb concludere, they are all concluda. Including tu removes the ambiguity and makes it perfectly clear that you are the one who has to finish the project.

I know in anticipo means early or ahead of time. What does l'anticipo mean here?

When used as a noun in a work or business context, l'anticipo translates to "the advance" or "advance payment". It shares the same core meaning of doing something ahead of time, but here it specifically refers to getting paid a portion of the money before the project is completed.