AT THE HEART of a thriving city -- and thus a critical item on the agenda of the next mayor -- is housing. The availability, affordability and quality of shelter matter not only as measurements of a local administration's management abilities but also as direct responses of government to the personal welfare of the people. There have been concrete improvements in both the supply and condition of housing throughout the city in the last four years. But as political challengers of Mayor Barry can note with authority, there are jobs yet undone, projects less than perfectly managed and ideas untested or overlooked. And always there are the harsh limitations of public money available for these purposes, no matter what a mayor's opponents may propose.

Just as official statistics are cited by challengers to point up increases in crime, an incumbent can come up with impressive figures on the housing supply. The mayor reports that more than 6,600 units have been constructed or rehabilitated, mostly for low-to-moderate-income families; nearly 3,000 public housing units have been renovated to meet code requirements and provide new housing for the elderly and handicapped.

Still, as Mrs. Harris, Mrs. Jarvis and Mr. Ray have observed, there are abandoned, dilapidated properties that could be returned to the market; there are permit procedures that impede construction of new units, and there are people living in substandard dwellings. Mrs. Harris, who has had more than a little experience in the field of public housing, also argues that available money has gone unspent because renovations and repairs have taken too long to get under way, that rent collections are poorly handled and that security has been inadequate.

Her proposals for improved management include budgeting that would place additional emphasis on early financial assessments project by project and new management personnel practices. These are welcome suggestions, as is the demand by Mrs. Harris for maximum, timely and cost-efficient use of every federal housing dollar available.

Lurking in the future of anyone who is mayor for the next four years is the cold fact that federal funds are not likely to pour forth in abundance. What money does flow should be used mainly to stimulate private investment in construction and rehabilitation projects similar to those undertaken during the last four years. As Mr. Barry has noted, the public/private partnerships can combine with the yields from tax-exempt bonds issued through the housing finance agency to provide still more low-to-moderate-income housing.

In addition to this balancing of money sources, types of shelter and project management, continued improvements in housing must be linked to economic development--to the new shopping centers, industrial activities and other job-producing projects that are either already scheduled or are included in the various campaign agendas. The development challenge will be the subject of further review in this space.