New York City Mayor Michael R. Bloomberg is expected to abandon his plan to ask public employees to pay more for their healthcare coverage when he unveils his latest budget proposal. The mayor had hoped to convince labor leaders that union members should pay 10% of their healthcare premiums, as part of an overall attempt to save $1 billion in pension and healthcare costs. Many employees now pay nothing, so a 10% contribution could save the city $350 million. But negotiations between the city and the unions have proved difficult.