cacher does, but cache as in “cache-toi !” (go hide!) and “je me cache” (I’m hiding) are pronounced “cash”.
Besides, “correct” pronunciation in a different language is pretty meaningless. The word may have come from French but we’re speaking English, not French.
Also, it might not be a loan word so much as a legacy-of-foreigners-taking-over word (c.f. the Normand invasion of Britain), which doesn’t tend to help the language’s users care about respecting the “original” pronunciation. I’m not certain when exactly cachet entered English.