Is my wage being illegally docked?

I recently did two days of temporary work through an agency at £25/hour, 8 hours per day (16 hours total).

Halfway through the second day (4 hours into the shift), I got a phone call from the recruiter saying the client wanted to decrease my pay. I said I’d think about it, and we agreed I’d give them an answer by the end of the day. At the end of the day, I told them I did not accept the new rate and that we’d part ways.

I'm requesting my pay. They are trying to pay me the lower rate for the whole of the second day, "as this was the rate which was decreased when you went back into work with them." They did not tell me they were even thinking about a new rate till halfway through the day.

I’m now requesting my pay. They’re trying to pay me the lower rate for the entire second day, claiming “this was the rate which was decreased when you went back into work with them.” They never mentioned any change until halfway through the day.

My question is:

1. Am I right in assuming they must pay me the originally agreed rate up until the moment I was told about the proposed change?

2. Do they also have to pay the original rate for the rest of that day, given that:

• I never agreed to the new rate,

• the change was raised halfway through the workday, not before I started,

• and I received nothing in writing about any change to pay or contract?

England