Yes, you need to contribute....depression and back problems? You've just described 90% of the work force.....ahem.
I have just moved in with my partner who is working 36 hours and I don't work due to depression and back problems. Am I entitled to help with money?
One word: YES.
Yes, you should help out financially, nobody gets to live for free.
Go get your own place and see how long you make it with no $
Get up, get a job, stand on your own two feet, and stop leaching off another.
I think the word you are looking for is "obligated" not entitled. You are not entitled to move in with someone and not contribute anything to running the household.
You are, however, obligated to help. You can get meds and therapy to help with depression; and your doctor can suggest exercises for your bad back.
I've had a bad back for years; now I have an office job where I don't have to do a lot of lifting. Call center work requires a pleasant voice and patience and no lifting. There are jobs that you can get that are not going to damage your back.
So, help your partner out and get busy helping. You'll feel better when you are contributing to the household, which will help with your depression.
You moved in without discussing the finances beforehand? Yea, this relationship is going to last. (eye roll) Honestly, if you are going to move in with someone there are ground rules that need to be discussed before anyone moves so much as a shirt into a closet. Finances and household chores are at the top of the list. Yes, you need to contribute financially. If you have no money do to so, than you need to make up for it in household up keep.