When applying for a new job when you already have one is it better to say you have a current job or to not specify? Would they be more likely to hire people that dont have jobs already?
I think you should always include your current job in your job application, for a few reasons:
You should always be as honest as possible. No companies want to hire people who hide the truth from them.
It lets them know that you will need to schedule interview time around a job.
It warns them that you'll likely have a notice period to work.
In my personal experience companies generally (depending on the type, my experience is in tech) quite like applications from people who are already employed. It can show that you are applying for the job because you like the job itself, rather than being unemployed and applying to every job available. Not saying this is the best way to think, but I've seen it a few times.
Prospective employers have a bias in favor of candidates who are currently working, so you are in luck. Assuming that you disclose that you are currently employed when requested to disclose, that is.
This disclosure is not usually done in the cover letter but within the body of whatever application form your prospective employer provides you.
You should include your current job so that you have something to talk about during the interview process that references your most recent experience. They will want to talk about what your current day to day workflow is like rather than from X years ago.
Also it gives you more negotiation power when talking salary, it's possible that companies will offer lower salaries to unemployed people.
Other answers have already listed several reasons to list your current job but I would like to add another one.
I think it's pretty well established that people who have been unemployed for some time have a very hard time getting a new job and that it only gets harder and harder.
There is even research showing that applications with a 6-12 months unemployment period at the end mostly get ignored, whereas nearly identical applications from applicants who left their previous jobs recently do get responses.
If you have been in your current job for more than a few months, not listing it is therefore simply not an option.