Should the quote marks after "focus" be removed?

Don't remove the quote marks. And replace the colon with a comma.

She said, "I think that’s very much going to be a focus."

When the quote ( by the same person) is more than one paragraph, the last sentence gets the final closing quote mark.

The first word in each quote paragraph has a quote before it.

So - yes, remove the quote after focus.

Also remove the last “she said”

I agree with this.

My impression was that there were two separate statements, not two connected paragraphs because they are so short. But yes, if these are paragraphs, PUNKEY is correct. These sorts of sentences taken out of context can be tricky to interpret.

I think this version works and see no reason to make it into two paragraphs:

Author of The New Royals, Katie told OK! that Charles and his wife are determined to address the issue of vulnerable sectors of society. She said, "I think that’s very much going to be a focus. We’ve seen that with Charles right from the outset – trying to help young people get on the ladder, get out of poverty or homelessness or drugs – and give them the start of a better life. His work with the Prince’s Trust will absolutely be his legacy."