Musical terminology isn't "Italian" - it's mostly conventional words and phrases that just look like they might be Italian.
In Italian, "andante" actually means "cheap" or "second-rate" - nothing to do with "walking" at all.
And "assai" doesn't always mean "much" or "a lot" - it can mean the opposite. A phrase like "m'importa assai" means "what do I care?" or "I don't give a **** about that".
Musical instructions written by a real Italian speaker can be just as confusing - like Handel's "largo e allegro" (not alternately - simultaneously!)
Not to mention classics like "Non si deve usare qui il maledetto legato d’organista da chiesa anglicana" ...
"Andante assai" can mean either slower or faster than "Andante" - but since we don't have a precise definition for the tempo of "Andante" anyway, you just have to make up your own mind, or blindly follow whatever MM mark that an editor added to the score.
One theory is that "andante" is an incorrect form of some part of the verb "andare" - but "andare" simply means "to go" with almost the same range of literal and metaphorical meanings as in English - you can certainly "andare a piedi" (i.e. "walk") but food can "andare a male" (go bad) or something can "andare perduto" (go missing).