Esa, sdp and taking in a lodger

If this is a commercial arrangement and the lodger is not a relative this would not affect the SDP.

You would lose your single person discount for Council Tax if you currently get it unless the lodger falls into a category of occupant that falls to be disregarded (such as students). This may not matter if your bill is covered for you by CTR but if you do have to pay a percentage of the bill that amount would increase.

Moreover the income from the lodger will be taken into account and deducted from your ESA less a disregard of £20/week.

You might find it helpful to read this
https://www.citizensadvice.org.uk/housing/renting-a-home/subletting-and-lodging/lodging/taking-in-a-lodger-what-you-need-to-think-about-first/

If you are going to have a lodger it would be prudent to have an agreement anyway.
For benefits purposes it will make things much simpler to prove the arrangement with an agreement.
i expect there are various templates available that you can find with an online search.

If you take in a lodger you will of course need to inform ESA as well as the council.