Nobody should take too much from the New York Times poll released last week. The poll was designed more to probe the minds of New Yorkers than to simulate a real election. But there are nuggets of information to be mined.

The next mayor must clear two hurdles. The first is his or her own party primary in September, and then victory in the general election two months later. Since primary voters tend to be the most committed to their party and also tend to be the most extreme, it could well be the fringes that set the tone. Democratic primary voters are generally to the left of the party as a whole, and Republican primary voters are more right wing.

This isn’t usually a problem for most Democratic candidates, since they enjoy a five-to-one advantage over Republicans. A Democrat doesn’t need a single Republican vote to win a citywide election. But it can be a problem for a Republican who must win Democratic votes to prevail in a hostile environment. That’s why the candidacy of Thomas Ognibene presents such a difficult challenge to the mayor. It is a test of the mayor’s ability to win the GOP nod without offending Democrats by moving too far to the right.

The Times poll shows how difficult this may be. Only 53% of Republican voters approve of the job Mr. Bloomberg is doing. Just before the last election, a Quinnipiac University poll found Mr. Bloomberg ahead of his Democratic opponent Mark Green among Republicans by 93% to 2%.

A more recent Quinnipiac poll raises more concerns for the mayor. It showed that party identification is very or somewhat important to 51% of Republicans. In 2001, when Mr. Bloomberg defeated another Democrat-turned-Republican, Herman Badillo, in the Republican primary, the issue of his Republican bona fides barely surfaced. But after three years of politics and policies that seem more like those of a liberal Democrat, the natives are getting restless.

The Times poll suggests that on traditional Republican issues, such as crime and taxes, it’s a mixed bag for the mayor.

On taxes, the poll shows that 54% of the respondents correctly recognize that taxes have increased since the mayor took office, and only 2% believe they have gone down. About 35% believe they have remained the same. This is potential trouble in a Republican primary. There’s better news with the numbers on another GOP concern, public safety.

Crime has continued to drop during the mayor’s tenure, a major plus for an incumbent seeking re-election. Some 35% of the respondents recognized this, while 45% erroneously believe the crime rate has remained the same. Worse, 17% wrongly believe crime has increased. This is clearly an issue the mayor needs to exploit. He has to get an unambiguous message across that crime continues to drop since he took office, and milk every vote from this good news. But, unfortunately, crime is no longer the uppermost concern. Education is.

Before the last election, 22% of the Times poll respondents thought that the next mayor needed to concentrate on crime above all issues, and 19% said education. Answering the same question earlier this month, 22% answered education and just 9% crime.

This is not good news for the mayor. The numbers on his handling of the schools are abysmal, with 68% of respondents indicating that they were dissatisfied with the quality of the public schools, a figure that increases to three-quarters among those identifying themselves as parents.

Evaluating the mayor’s assumption of control over education, 48% of those surveyed now think it was a bad idea, as opposed to 39% who believe it was a good idea.

The last time a Republican mayor lost his party’s nomination was in 1969, when Mayor Lindsay lost to a conservative Republican state senator, John Marchi of Staten Island. Lindsay prevailed in the general election, running on the now-defunct Liberal Party line, because his Democratic opponent, Mario Procaccino, was far more conservative than Lindsay, during the heyday of liberal influence. A parade of liberal Democrats jumped onto the Lindsay bandwagon.

Mr. Procaccino won the Democratic primary because four more liberal opponents divided two-thirds of the vote, enabling the conservative city comptroller to win his party’s nod. As a result, the runoff system was put in place for citywide offices, establishing a 40% threshold to win a party’s nomination.

If Mr. Bloomberg loses the Republican line to Mr. Ognibene, who will also presumably win the Conservative line, can Mr. Bloomberg beat a liberal Democrat (and all four of the Democratic hopefuls are bona-fide liberals) by staking out middle ground?

The concern over the Republican primary is now on the front burner at City Hall. Late last week, I reported that Bronx Republicans wanted to hold off on a mayoral endorsement until after former state Senator Guy Velella was released from jail. Even as these words were penned, Bloomberg operatives were doing an end-run around the self-described “interim” Bronx GOP chief, Victor Tosi. Going directly to the district leaders, they forced Mr. Tosi to hold a vote on Saturday. Seeing the writing on the wall, the anti-Bloomberg forces caved in.