ESA to UC migration due to HB increase?

I've been on legacy benefits - ESA (support group) - and ADP - for several years. I've also lived in the same 1-bed private rented flat for several years.

I have a query regarding the possible effects of a potential change in my Housing Benefit amount.

In the last couple of years the rent on my flat has increased several times and it is now significantly higher than the amount at which it was originally set.

However, I've just realised that my Housing Benefit has not increased at all since I first moved in. As a result, having checked the updated Local Housing Allowance rate for this financial year, I can see that the amount I am receiving is some way under my current entitlement. It seems clear that I should advise the Council of the change in my rent and request an increase in the amount I am paid.

My questions are:

1. What is the best way to do this, and what information do I need to provide - do I need to get and submit an updated tenancy agreement from my landlord, or will a letter from him suffice?

2. Is there any chance that requesting/receiving an increase in HB amount may trigger an ESA or ADP review, or a migration to UC?

Thanks in advance.

Thanks David - do you have an answer to the second question, regarding the possibility of an HB amount change causing a migration to UC?