Yes, there is a fine. If the city has chosen to enforce it, it would be up to nearly $3000 by now.

Sec. 27-32. - Ice, snow, sleet—Duty of abutting owner to remove or abate.
(a)
Removal. The owner of land or a building, bridge or premises fronting upon the sidewalk, or connected with a footway, shall remove snow, sleet or ice within a minimum of forty-two (42) inches of such property including handicap ramps within twenty-four (24) hours after a storm ends.
(b)
Abatement. If the owner of the property referred to in subsection (a) cannot entirely remove the snow, sleet or ice within the aforesaid parameters, he/she shall cover the affected portion(s) with an abrasive substance so as to keep the property in a reasonably safe condition at all times.
(c)
Penalty. If a property owner fails to comply with subsections (a) and (b), above, within twenty-four (24) hours after receiving notice to remove the snow, sleet or ice from an appropriate city official, said owner shall be punished by a one hundred dollar ($100.00) fine, or an amount up to the maximum allowed under state statute. Each day that the violation continues shall constitute a separate offense.
(Code of 1985, § 27-9; Ord. of 10-17-88, § 16 Ord. No. 1426, 9-28-06; Ord. No. 1587, 4-20-09)

https://www.municode.com/library/ct/new_haven/codes/code_of_ordinances?nodeId=TITIIICOGEOR_CH27STSIPUWA_ARTIINUOBDACO_S27-32ICSNSLUTABOWREAB

Who can we approach about this? Is it LCI?

I walk this area every day. Power gas has really made zero effort in maintaining the sidewalks around their property. They are completely ice and snow covered and have large piles at the ends and corner. The empty threat of city fines does nothing, I have stopped making any sort of purchases there, I would suggest others do too.

You could try contacting your LCI neighborhood representative. Here's the directory:
http://www.cityofnewhaven.com/LCI/Administration/Staff.asp

You could also try contacting your elected officials. They should at least know who to contact. I wrote to my alder who then wrote to DPW about snow removal. I haven't heard back from the mayor's office.

Email the Mayor at MayorHarp@newhavenct.net
Find your Alder's email: http://www.cityofnewhaven.com/Aldermen/index.asp

I just emailed our Alder about this issue and the H& R Block issue. Our Alder is Abigail Roth.

We are closing out-dated snow related issues in order to more efficiently manage new posts. Thank you.