The only track from Holly and the Italians that I remember from my youth is 1980's Miles Away:
Looking back, this should have been some sort of hit: it's a perfect power-pop song, and Holly Beth Vincent's voice has star-quality (w/ Florence Welch esp. in Dog Days perhaps being the closest comparison case), and she's flat out cool in the (gold-standard) Chrissie Hynde/PJ Harvey-ish/Karen O-ish way in the vid.. Mark Sidgwick and Steve Dalton (a.k.a. Steve Young) monster the bass and drums respectively on this one like they're McKagan and Adler respectively. (Were the latter guys listening?)
Other tracks seem specifically influential on others:
Youth Coup sounds like both Liz Phair's Exile (esp. 6'1") and PJ Harvey's Stories (esp. Good Fortune) to me.
Rock against Romance sounds a lot like G'n'R's Sweet Child o' Mine (only without Slash's great signature riff and Axel's special yowl over the top).
HATI evidently cut it live too: here are stonking versions of Youth Coup and Rock against Romance on OGWT (live in the tv studio w/o an audience). Holly's skinny sleek and is on her guitar (just rocking basically), injured leg notwithstanding:
Or consider Poster Boy:
I think it substantially anticipates both G'n'R's Rocket Queen and Green Day's When I Come Around.
And consider Means to a Den:
Surely that anticipates much of C86 pop (and Juliana Hatfield and...).
HATI also did quite a nice line in more traditional girl-pop, including esp. a cover of the Chiffons' Just for Tonight (for which the great Ellie Greenwich did some backing vox) as well as the single that got them a record deal in the first place, Tell That Girl to Shut Up.
In sum, HATI was a very good band with a magnetic singer and front-woman that never quite got its due, except indirectly/sincerely, by being diffusely influential and possibly being extensively strip-mined by others.