Final answer:
The correct term for the given context is 'sync', reflecting a state of synchronization or harmony. Although 'synch' is an alternative spelling, 'sync' is the more common and preferred abbreviation in modern usage.
Step-by-step explanation:
The question regarding whether the correct term is 'synch' or 'sync' relates to English vocabulary and spelling. The correct spelling in this context is 'sync', which is a common abbreviation of 'synchronization'. The term 'synch' is an alternative spelling but is less commonly used. In the given sentence, 'It was not the first time that the mayor has seemed out of sync with the people he leads', 'sync' is the appropriate choice because it reflects the mayor's lack of harmony or alignment with the sentiments of the public.