Can I use free 144 hour visa.

Travelling on Ist October UK To Shanghai, arriving on 2nd, 3 hour wait, then flight to Xian.
2 nights Xian, Then bullet train to Beijing, leaving on 7th on a Cruise. No return to mainland other than day in Hong Kong, and 1night in Singapore before flying home.
Very complicated trying to sort out whether a full entry permit is required or not. Thanks.