In general, yes, but not if the employee holds a 'politically restricted post', as defined by Section 2 of the Local Government and Housing Act 1989. This section defines a list of posts within a local authority which are always politically restricted:
The following persons are to be regarded for the purposes of this Part as holding politically restricted ...
Not at the time that this question was posted - but a poll was conducted between October 22nd and 23rd by Emerson College/PIX11/NewsNation which gave Eric Adams, the Democratic Party candidate, a 40-point lead over his Republican opponent, Curtis Sliwa.
The Emerson College/PIX11/NewsNation poll of the New York City mayoral
race finds Democrat Eric Adams ...
Tl:dr According to one website, no polls have been released yet.
According to Crowd Wisdom 360
no NYC Mayoral Polls 2021 have been released yet
Disclaimer: I couldn’t find a fact check for this website, so it might be untrustworthy or false.