There is a simple way of expressing your "desperation" without having to invoke emotions. But first let's have a brief definition of it in context:
Desperation is the situation, in which one is in strong, timely need for a job in order to maintain his/her life style.
So just to clarify, there is both, an emotional and rational component to it. It is your situation which renders you desperate, and it is your emotions which recognize and express that.
It is true that expressing desperation indicates to employers that they can "exploit" you by paying a lower wage. However, if you are desperate, you would want to work for a lower wage. You can get that job by demanding less money, which makes you more powerful in the negotiation process. If you perform well enough, you should get a raise based on your actual qualifications minus the desperation after the probation time expires. Obviously you have to negotiate again for it - but if you feel that they helped you, you should at least mention that you are thankful for having employed you. Employers are humans after all, and value positive character attributes and honesty (not all obviously). It's obviously preferable to toxic personalities, as having conflicts within a business can result in terrible outcomes, up to having to fire people and demoralizing employees (and possibly the employer as well).
So: Simply show your "desperation" by demanding a lower wage upon application than what is common, without begging for the job in any shape or form. Give them time to prove your worth and renegotiate over time once you are in. Understanding this is a mark of the ability to plan in the long-term and therefore intelligence (or wisdom). That is likely to be positively recognized in an application.
Employers usually can see through applicants, so you shouldn't try to cover up what is obvious (if it is), but instead gracefully recognize it and show it in your wage negotiation. It is important that you acknowledge your situation and mention it at least, so that the employer doesn't get other ideas to why you negotiate less than what you (your work) may be actually worth. Find a good phrasing for it of course.
Having a fresh employee for a lower wage for the probation time, who in turn shows potential to be good in the mid or long-term, is a valuable opportunity for businesses. It is much better than paying someone the "full" wage, who in turn turns out to be a bad and expensive employee within the probation time.