Debating its £372m budget for the year, the ruling Labour cabinet said its proposals would "make a difference".
Plans to close Marksbury Road library in Bedminster and cut £160,000 of funding to adult learning courses have been scrapped after local opposition.
Final recommendations will be debated at full council on 26 February.
Nearly £1.5m would be spent on recruiting an extra 42 Police Community Support Officers.
Efficiency savings
A further £2.4m would go towards offering older and disabled people live independently.
A new rape crisis centre would open in the city, costing £75,000 a year.
Money is also being spent on initiatives to combat fly-tipping and cleaning up neighbourhoods.
The new investments would be paid for mainly through efficiency savings elsewhere in the budget.
Council leader Helen Holland said: "This is an exciting budget that will deliver safer streets, healthier communities, better neighbourhoods and greater equality and justice.
"It will make a real difference to the way we deliver services, particularly to the most vulnerable people in our city."
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