Dear @John1111,
You need to inform the Jobcenter in the place you wish to move to. They will examine your reasons for wanting to move and whether the costs of your new accomodation are ok. They only approve for certain reasons, such as taking up a job in a different city. The new Jobcenter is also the one you need to ask if you wish for them to loan you money to pay a deposit on a new place.
If you wish for help with moving costs, then you ask for this with your current Jobcenter.
If you move without getting the approval of Jobcenter first, then they will not assist you with any moving costs or paying the deposit. They will also at least at first not give you additional money for rent if the rent of the new place is more expensive. They will just continue to pay the old rent amount if your new rent is more.
Here is some extra information about moving when receiving support from the Jobcenter:
https://hartz4widerspruch-de.translate.goog/ratgeber/wohnen/umzugskosten/?_x_tr_sl=de&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=wapp#
Best,
Éanna