Hiring takes several forms. The standard is take a civil service exam and get appointed (what they call “called off a list”).
City jobs are always listed on the official NYC website. Sorry, no links. You can start your search there. Once you’re in the listings, you’ll see that it has a lot of filters such as managerial, FT, PT, locations etc.
Being hired as a provisional means, while you get the same civil service benefits, you’re not protected in the event of layoffs in the same way civil service employees are. But layoffs haven’t happened in a very long time, so I wouldn’t make that a criteria. BTW, new hires are enrolled in a Tier 6 pension plan which is a long distance from the famous NYC pensions our parents generation had. Also, the teacher’s union has a better pension as well. But the city does offer a pre-tax deferred comp option, which is a 457 plan.
At the upper managerial levels you’re more likely to find the political appointees. But there often is a lot of turnover at that level every time we get a new mayor. But if you’re that “connected”, you should be able to land on your feet if you are laid off.