New York Mayor Edward I. Koch (D) has had a troubled third term, what with city government scandals and the racial crisis at Howard Beach, and his poll ratings reflect the voters' concern. A Gallup poll taken last month showed his job approval rating at 50 percent, its lowest point. The irrepressible Koch said he was "surprised it was so high."

"After something like 15, 16 months of constant attacks on the administration, that I would have a 50 percent favorable rating, I think that's extraordinary," Koch said at a news conference. "I think it's terrific."

Koch won reelection in 1985 with 78 percent of the vote. But 45 percent of those surveyed in June said they do not want the mayor to run for another term.

Meanwhile, on the West coast, Los Angeles Mayor Tom Bradley (D), receives a 65 percent job approval rating, according to a Los Angeles Times survey, and leads all potential challengers, should he seek a fourth term in two years. But the news is not all good for Bradley. Only 31 percent of those polled think the mayor and the city are "in step," and 45 percent said Los Angeles is changing faster than the mayor.