If your local authority has a Welfare Rights Dept they will be able to confirm. If you receive benefits and need support to care for you that person can stay 3 nights per week (without it affecting your benefits), in reality this needs to be declared though. If somebody lives in your porperty the Council will take that into account and that person has to contribute (and in reality will make a difference to) to the rent and council tax. Regardles of whether that person has an income. You say your ex partner is too proud to sign on, where are they living now and who is financing them. I am really sorry that this factual information is probably not what you need to hear just now but as you say you can do without the additional stress as you've got yourself and you son to think about.
Take Care xx
I have been following your posts this week and felt the need to reply to this one, my niece is in a similar situation and was told that she could have her partner live with her 3 nights prin week without it affecting her benefits, but that was with her boyfriend working - thus maybe different if your child's father is out of work. I think you can speak to the benefits office for advice xxxx