Ultimately, it has a lot more to do with his position and resources in the election than anything about him personally. Although his opponents would have you believe the mayor is "scared" to debate the issues or "doesn't believe in the democratic process," this is hardly the case. Debates are certainly nice forums for citizens to learn about candidates and issues, but they are also key political moments for campaigns. If a debate is more helpful than hurtful for your campaign, you want it. If it isn't, you don't. Very simple, very machiavelian. Five points:
Point #1 - Frontrunners hate debates / chasers love debates - If you watched The West Wing last night, this was abundently clear. Congressman Santos, trailing by 9 points in the polls, was begging for a debate, but his political handlers knew that Senator Vinnick had no interest in debating, since he was already ahead. As Vinnick's handler says, "They need to debate 10 times more than we do." The bottom line is that if you are ahead in a campaign, you want to minimize "big moments," where you can either squander your lead with a dumb mistake or your opponent can gain ground with a brilliant manuever. Debates are inherently "big moments." Therefore, people in the lead hate them and people who are behind beg for them. Jennings is in the lead. His opponents are far behind.
Point #2 - Well-funded campaigns hate debates / poor campaings love debates - Debates are the best advertising at the cheapest cost - you get the full attention of interested voters at absolutely no cost, and you get to say whatever you want. This means your message will be up to date. Jennings has plenty of money and can run all the paid ads he wants. His opponents are cash-poor and would love some free airtime. Therefore, Jennings doesn't want to debate and his opponents do.
Point #3 - Mainstream candidates hate debates / marginal candidates love debates - a debate has a strange leveling power in the minds of low-information voters. If a low-information voter sees three candidates in a debate, he/she accepts that these are the "legitimate" candidates in the election. This is why the major parties try so hard to keep third party candidates - like Perot in '92 and Nader in '00 - out of the debates. It denies them the legitimacy that comes with being a "big boy" in the debate. If you can't keep your marginal opponents out of the debate, you simply try not to debate them. Like it or not, Alice Green and Joe Sullivan are marginal candidates. Thus, Jennings doesn't want to debate them.
Point #4 - Incumbents hate debates / Challengers love debates - When there is an incumbent running for office, debates generally center around his/her policies. Since challengers are not managing policy themselves, they often have little record to critique. This usually means that debates end up being asymetric affairs, with challengers able to attack the incumbents record and the incumbent left to either defend his record or attack the challengers experience. Except in the best of situations, this works to the advantage of the challenger. Challengers also tend to be less "legitimate" than incumbents, and thus gain ground under point #3. Thus, Jennings doesn't want to debate and his opponents do.
Point #5 - Not debating can hurt you, but only in high visibility, close campaings - Ducking debates against serious opponents, for instance if you were running for president of the United States, would seriously hurt your polling numbers. But voters tend to not be well aware of the day-to-day news of mayoral campaigns, so they don't usually punish debate duckers. Ducking a debate against a marginal candidate is often less problematic than against a serious candidate in a close race, because voters expect debates in close races. Thus, Jennings doesn't mind missing the debate.
As you can see, Jennings has no reason to debate. His opponents have every reason to demand one. They will, and he won't. Case closed. Jennings is a hack, but right now he's just being a smart politician. His opponents will try to goad him into a debate using every rhetorical device in the book, but they know he won't fall for it. They need the debate, he doesn't.
Debates are wonderful traditions in democratic communities. They inform voters, allowing for the airing of issues, and let voters see politicians interact with each other. But for campaigns, they are either vital chances to gain ground or briar patches to be avoided at all costs. And they are treated as such by the candidates, through pure political calculations about the benefits and drawbacks that attending the debate has for their chances in the race. No more, no less. It has nothing to do with anything personal about the candidates. Joe Sullivan and Alice Green would be stonewalling just the same if they were in Jennings position. They might come up with better excuses - Jennings is laughably claiming he's "too busy" - but they would decline just the same. But that's the sad reality of debates. To understand them through the lens of the democratic ideal is just that - idealist. Enough already.
P.S. - It can be the case that everyone wants a debate. This happens often in wide-open races when no one is really sure what is going to happen, like the Soares/Clyne/Cusick race for county DA last fall (the general election, not the primary). Soares was the frontrunner, but needed the debate to silence his doubters. Clyne and Cusick both needed to make up ground. It can't produce a positive-sum result, but it can be percieved as a positive opportunity by everyone heading into the event.
P.P.S - I wish I could put the TU cartoon up, but it's not available on the website! I guess I'll just have to