Learn Arabic (Modern Standard): يتمنى أن يصبح مهندسا.

The word أن (an) connects two verbs, similar to "to" in English ("wishes to become"). In Arabic, it also changes the pronunciation of the second verb's ending. It gives يصبح (he becomes) an "a" sound (fatha) at the end, making it يصبحَ (yusbiha) instead of its normal default ending يصبحُ (yusbihu).

Why is there an extra alif at the end of مهندسا (engineer)?

That extra alif carries the "an" sound (tanwin fatha) to show the word is the target or "predicate" of the verb يصبح (to become). Verbs like "to be" and "to become" make the noun that follows them take this specific ending (the accusative case). So مهندس (muhandis) becomes مهندساً (muhandisan).

Why does يتمنى end with a ى instead of a regular ا or ي?

That letter is called an alif maqsurah (ى). It looks like a ي (yaa) without the dots, but it is pronounced exactly like a regular ا (alif) — a long "aa" sound. It is simply a special spelling used at the end of certain Arabic words. The verb is pronounced "yatamannaa".

Where is the word for "he" in this sentence?

The "he" is built directly into the verbs! Both يتمنى (he wishes) and يصبح (he becomes) start with the prefix يـ (ya-), which marks the "he" form in the present tense. In Arabic, you do not need to add the separate pronoun هو (he) unless you want to add special emphasis.