A registered office is the statutory address every UK incorporated company must maintain at Companies House, where official documents can be delivered. A process agent is appointed by contract to accept service for a party that has no UK presence at all. If your company is incorporated overseas, a registered office is not something you have here, which is precisely why the agreement asks for an agent.
What a registered office does
Companies formed in the UK must keep a registered office address on the public register, and statutory mail and legal documents can be delivered to a UK company there. It exists because the company itself exists here. An overseas company with no UK establishment has no registered office in this jurisdiction, and nothing at Companies House creates one for it.
Where the process agent fits
The process agent fills exactly that gap for the purposes of a particular agreement. Rather than the company having a standing statutory address, the contract names an agent in England and Wales at whose address documents relating to that agreement can be validly served. The appointment is private, transaction specific and time limited to the term you choose, from £125 per year with Tremark Process Agents.
A common point of confusion
Overseas groups with a UK subsidiary sometimes assume the subsidiary's registered office can simply absorb the role. It can be named as agent if the counterparty accepts it, but that is a contractual appointment like any other, with all the continuity risks we describe in using your own director or subsidiary. The registered office status itself does nothing for a contract signed by the overseas parent.