Mayor Jeannette McCarthy is again asking the City Council for funds to conduct an independent investigation of the events leading up to the arrest of Police Chief Thomas LaCroix in Maynard last summer. McCarthy’s request for $10,000, to be appropriated from a Personnel Department account, is on the agenda for the council’s 8 p.m. meeting Monday in City Hall. Earlier this month, councilors balked at the mayor’s request for $20,000 from the city’s free cash account for an internal investigation, which would help find out whether LaCroix, who was placed on paid administrative leave after the incident, had broken any departmental rules. Waltham police officials said there is an inherent conflict of interest in investigating a superior-ranking officer, leading to the request for an independent consultant. Councilors questioned the timing of the request, and tabled the matter after a long discussion on Feb. 4. LaCroix is facing charges that he allegedly assaulted his wife and a female neighbor during separate incidents in June, leading to his arrest by Maynard police. In her recent e-mailed request to the City Council, McCarthy said she wants the city to pursue the investigation promptly after the conclusion of LaCroix’s trial, which is slated to begin March 27, so that a decision can be reached quickly on his status as head of the Police Department. During LaCroix’s suspension, Keith MacPherson has been serving as acting police chief.