Learn French: Est-ce que tu as eu le temps récemment ?

This is the Passé Composé (past tense) of the verb avoir. To say 'you had' or 'you have had', you need the helping verb (tu as) plus the irregular past participle of avoir, which is eu. Together, tu as eu creates the past tense.

How do you pronounce the past participle eu?

It looks tricky, but it is pronounced exactly like the French letter U. It sounds just like the vowel sound in the word tu.

Why is it le temps instead of just temps? In English we just ask if someone 'had time'.

French almost always requires an article before a noun. While English drops it in the general phrase 'to have time', French uses the set phrase avoir le temps (literally 'to have the time').

Do I have to use Est-ce que to ask this question?

Not at all! You could also invert the subject and verb to ask As-tu eu le temps récemment ? But adding Est-ce que to the front of a normal sentence is a very common, natural way to turn it into a yes/no question in everyday French.