Learn Arabic (Modern Standard): يمشي أبي كل صباح بهدف تحسين صحته. - My father walks every morning with the aim of improving his health.

In this sentence, كل صباح acts as an adverb of time showing when the walking happens. Adverbs of time and place in Arabic are generally in the accusative case (mansub), so كل takes a fatha on its final letter.

Is هدف تحسين صحته a single grammatical structure?

Yes, this is a nested idafa (genitive or possessive construction). It literally chains three nouns together: the aim of (the) improving of (the) health of him. Because of this chain, both تحسين and صحة are in the genitive case (majrur) and take kasras.

How do I pronounce the attached pronoun ـه at the very end of the sentence?

It is pronounced as "-hi" instead of the default "-hu". Because صحة is in the genitive case from the idafa chain, its final letter has a kasra. When the "him/his" pronoun follows a kasra or a "yaa", it shifts from "-hu" to "-hi" for smoother pronunciation, resulting in صحّتِهِ (siḥḥatihi).

I know تحسين means "improving", but how can I quickly identify its root verb?

The word تحسين follows the strict pattern تفعيل (taf'eel). This is the standard template for the verbal noun (masdar) of a Form II verb. Whenever you see this pattern, you know it comes from a verb with a doubled middle letter, which in this case is حسّن (to improve).