How should I explain a previous refusal in a new UK visa application?

The removal notice is here...

After my girlfriend's visa being cancelled at UK border, will her UK visa application be banned?

To start, let's get the terminology right. Your gf was not 'rejected' she was 'removed' from the UK because she made false representations when she applied for a visa. The removal notice indicates that the IO initially got upset because your gf was playing cat-and-mouse about having a bf in the UK and this led to a secondary level of questioning.

She wrote on the application that she contemplated a short visit to see family and friends. That's fine. Then she changed it to 4 months and got bounced. You propose to paint this incident as an innocent mistake arising from her understanding of the rules. Bzzzt. Wrong. That would be an even bigger mistake, here's why...

At the application stage your gf portrayed herself as a standard visitor; she went as far as to fabricate an existing family in the UK. She succeeded in painting them a picture and they accepted her premise for a short visit. The picture was the type of person who can make visits to the UK within the confines of their lifestyle and their ties to their home country. A person who wants to make a visit for 4 months clearly does not have the same lifestyle and home country ties as a person who wants to come for 14 days. Both types can easily qualify as long as they present a credible premise.

When she landed, she adopted a different profile, the profile of a person who stays for a long time.

This is not to say that the length of one's visit is seen more positively if they write down a very short period of time. It means the visit itinerary, lifestyle, duration, ties, and so on need to fit into one sensible package, or else it risks being 'incoherent'. Your gf depicted herself as a standard visitor, but her real agenda was to take up a trial relationship with you. Why didn't she come out and say it?

I fell in love with this guy and we want to have a trial relationship within the confines and stipulations of Appendix v of the rules (I have a print-out of Appendix V right here). We do not intend to establish a permanent household. We believe a maximum of four months is a reasonable amount of time for a trial relationship (given that it may end earlier). In all events I will return to XXXX well before any performance issues arise.

Instead she played cat-and-mouse with the IO (that's a very bad thing, I cannot stress enough what a horrible strategy that is) and they ultimately bounced her. A fair catch based on the available info.

It's very difficult for visa officials to believe this was an innocent mistake. People with ties to their home country cannot extend their visit 8-fold on a whim, instead they have commitments and a lifestyle that requires them to return as originally planned. It's also generally true that genuine hosts in the UK do not just extend their offer by 8-fold also because it means they are going from having a guest to having a lodger. Because of these things, they are more likely to believe that she had previously arranged with you to stay for a long time.

TL;DR

You need to think up some other reason than putting it all down to an innocent mistake. Regardless of the circumstances, in their eyes it will be an attempt to cover up a lie with yet another lie.

User Michael Hampton (to whom thanks) has pointed out in comments that you should consider instructing a professional to craft an explanation that the visa officers will find satisfactory. You can use ILPA to search and send a few choices a print-out of this answer and your earlier answer for background. She does not qualify for legal aid and you/she should understand that this type of service attracts a fee.

Random Notes:

• Starting this year and in concert with the UK exit controls bedded down, they are actually starting to refuse entry clearance applications where the person stayed significantly longer than they said. This site has not yet had a refusal on those grounds, but I have the refusal formulae and practitioners are seeing them.

• British Immigration Officers are not stupid; they know cat-and-mouse games when they see it happening. Unless you are somebody like Saul Goodman, the best option is to play it straight up with candour and transparency.

• So why is the Standard Visitor Visa good for 6 months if using that amount of time will raise eyebrows? In the OP's case it's not about the 6 months, it's about what she wrote down on the form and the picture she painted for them. All the rest is a different question. Indeed, a fair question, but a different question.

• This answer seems to indicate that it's against the rules to have a change of circumstances and stay longer than what was intended. Is it against the rules? No. It's against the rules to make false representations.

• From user pnuts: "I'm looking for official documented guidance about what's expected to be put into an application form to explain any previous rejection." The form comes with instructions and if a specific area is not in the instructions, a person is supposed to use the natural meaning of the words and answer the questions honestly. Outside of that "guidance" there is no 'official documented guidance'. There is an abundance of stuff about it, but it's behind paywalls, or in references like "Macdonalds".