The release of the preliminary mayor’s management report yesterday gave a sharp statistical snapshot of the challenges Mayor Bloomberg faces in his second term, including a four-year high school graduation rate of 53.2% and a climbing failure rate in responding within 24 hours to reports of child abuse.

According to data released by the mayor’s office yesterday, the city saw a good beginning to fiscal year 2006 in many areas. The murder rate was at a record low, fewer children were poisoned by lead paint, the welfare rolls thinned out, more construction permits were issued, syphilis infections dropped for the first time in four years, and quality-of-life indicators like cleaner streets improved across the board.

“I think that we’re off to a terrific start, and performance remained strong during the early part of the year, and we continue to make progress in key areas,” the director of the mayor’s office of operations, Susan Kupferman, said.

But mixed in with all the positive statistics – which the mayor has said show that the city is “moving in the right direction” – are some numbers that are not moving in the direction city officials say they are aiming for.

The percentage of high school students graduating in four years dipped to 53.2% in fiscal year 2005 from 54.3% the year before, and school attendance rates declined slightly for the lower grades.

The city’s graduation rate became a sore spot for Mr. Bloomberg during his re-election campaign when his Democratic challenger, Fernando Ferrer, cited it as a failing of the Bloomberg administration. At the time, Mr. Bloomberg said Mr. Ferrer was using inaccurate figures.

This year, the Bloomberg administration added a new category to its preliminary mayor’s management report, pooling the number of students who graduated in four years and those who were still enrolled for a fifth. That category showed slight improvement, with the rates jumping to 84.8% in 2005 from 83.7% the year before.

The Department of Education’s senior instructional manager for assessment and accountability, Lori Mei, said many of the students enrolled for a fifth year are on track to graduate. She also said that while the decline in the four-year graduation rate is “disturbing,” there are only 527 fewer students graduating in four years.

The president of the United Federation of Teachers, Randi Weingarten, called the graduation rate dip “a sobering reminder that we are not making the headway we need.”

The city’s beleaguered child welfare agency has been under a magnifying glass since last month, when a 7-year-old girl was found beaten to death in her Brooklyn apartment.

The agency slipped when it came to responding within 24 hours to reports of abuse or neglect. Between July and October 2005, caseworkers at the Administration for Children’s Services responded within a day to those reports 93.9% of the time, down from 98% in the same period the year before.

A spokeswoman for ACS, Tanya Valle-Batista, said that while the agency is concerned about the change, it holds itself to a higher standard than most jurisdictions and is hiring 525 new caseworkers to spur improvement. The mayor has earmarked about $16 million to beef up the agency.

New Yorkers apparently overwhelmed the Department of Finance with e-mail inquiries, as the agency’s average response time to e-mail skyrocketed from just over three days to a full month.

The city was targeted by 30% more lawsuits than in the same period last year, more than 3,000 in four months, a jump the Law Department attributed to an influx of filings related to the World Trade Center attacks from people who chose not to seek payment from the Federal Victim’s Compensation Fund.

The city also reached deeper into its pockets for lawsuits, making twice as many tort payouts over $1 million as last year. And, according to the report, New Yorkers continued to flout the law and chat on their cell phones while driving. Police officers caught more of them: summonses for cell phone violations increased by about 20%.

Response times to fires jumped by 14 seconds citywide and dropped only on Staten Island, where an additional firehouse recently opened. Fire Department officials attributed the overall increase, in part, to a 10% increase in fire incidents. Fire-related deaths also increased to 22 from last year’s record low of 14.They remained lower than in 2004.

The Fire Department completed 35% fewer field inspections than last year. The report pointed to a more extensive inspection process, as well as inclement weather, as possible causes of the drop.

The 311 government hotline Mr. Bloomberg launched during his first term saw an 11% spike in call volume in the first four months of this fiscal year compared with the same window last year and about 4.2 million calls during that time. Non-English speakers, particularly those who speak Russian as a first language, placed 56% more calls. The top topic was noise, rather than the property tax rebate people were calling to ask questions about before. At the end of last year, after more than a year of haggling with the City Council, the mayor signed into law a new noise code that will regulate everything from jackhammer drilling to garbage pickups.

Meanwhile, number of rodent complaints rose, as did the number of food establishments failing health inspections.