No they won't. Unless you need to downsize, that might help. If you were in private rental you'd get more points for unaffordability but not increased banding I don't think. As you're already in HA I don't think it'll make any difference. You'll need to prioritise the rent above everything else so you don't get into arrears.
Check your tenancy, you may be allowed to sublet to a lodger, assuming you have a spare bedroom. This income could impact any benefits you're on though, including if you're getting the single adult council tax discount.
Have you checked on "entitled to" website to see if you're getting everything you should be? You can also input hypothetical circumstances to see how getting a lodger might affect you.
The lodger doesn't gain rights to the tenancy of your HA property. Lodgers don't have many rights at all. You set the rules, they must comply. Notice period of 1 week on either side to leave.
They'll need a kind of tenancy agreement, "licence to occupy" since they're a lodger not a tenant. You can probably download something for free, used to be able to buy them cheap in WHSmith.
Any trouble with them you can ask them to leave and change the locks if necessary. You don't have to do an eviction process through the courts like you do with a tenancy, so no risk of them staying longer than you want or without paying etc. You also don't need a "good enough" reason to ask them to leave, a personality clash or you wanting your home back solely to yourself is enough of a reason.
Failing any of that you'll have to look at reducing your expenditure. Or increasing your wages viaore hours or a better paid job. If you want to post your annual budget I'm happy to help you cull the stuff you could live without.