Here, nuit is the conjugated form of the verb nuire, which means “to hurt”.
The gloss for your sentence would run something like this:
Cet automate de mari lui nuit plus qu’il (ne lui) sert
This automaton of (a) husband harms her more than he serves
lui is a clitic pronoun standing for the indirect object of the verb nuire (‘to harm’). Here lui refers to the wife and nuit is the 3SG present indicative of nuire.
“harm” implies physical damage too much for my tastes. It’d say something like “gives her more trouble than he helps.”
In addition to those given in previous answers, here are several other idiomatic, albeit less literal, ways to translate “lui nuit plus qu’il ne lui sert” that might fit your context:
“… causes {her} more problems/creates more problems {for her} than he
solves”
(from ‘The Rebel Wife’ By Donna Dalton via Google Books);“… does {her} more harm than
good”
(from ‘Temple bar, conducted by G.A. Sala’ By Temple almanac via
Google Books); and“… is more trouble than he’s
worth {to her}”
(from ‘AN Unthymely Death’ By Susan Wittig Albert via
Google Books)
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