After the collapse of Austria-Hungary towards the end of WW1, Admiral Miklos Horthy took charge of Hungary as its regent, ostensibly ruling it for the Habsburgs in their absence. We know that the Habsburgs never regained control of Austria or Hungary (and there seemed to be little interest in restoring Kaiser Karl to the throne), but Hungary continued to be ruled as a regency.
How did this work, exactly? Was there any understanding among the people that the country might pass back into the hands of the Habsburgs, or was this merely lipservice paid to legitimize Horthy's position? Why adopt this position at all?
ostensibly ruling it for the Habsburgs
The Entente declared, that no Habsburg can sit on the Hungarian throne, else they will send in armies. Horthy worked for the Habsburgs before the war, he stayed a sympathizer, even after he declared that the Hungarian government wont support the Habsburg claims on the throne in march 1921.
and there seemed to be little interest in restoring Kaiser Karl to the throne
The Habsburg house disclaimed the thrones only in 1966.
After the fall of A-H revolution broke out in the Empire. Hungary had more revolutionary governments between 1919 and 1920 (republic, soviet republic, republic again and the kingdom without a king), they had different goals, but their main activity was to terrorize the other parties.
The main issue of the Entente was that none of these governments wanted to sign the Trianon peace treaty, some of them even wanted to restore Habsburg rulership. The Entente didnt wanted the Habsburgs, but they "offered" their candidate: Hungary can be a Kingdom, with the personal union of the Romanian and the Habsburg house, with the Romanian king as the ruler. It was refused by the Habsburgs.
The Entente desperately needded someone who they can trust. Horthy was the last Commander of the A-H fleet, and he already prooved himself, when he surrendered the fleet upon the King's command.
During the Romanian occupation and sacking of Hungary, Horthy was appointed Supreme Commander of the Hungarian armies (at that point there was no such thing as a Hungarian army, but Horthy promised, if the Entente allows it, he will raise an army in 4 days and will restore order).
After the occupiers left the country, Horthy had the only army in the country, and he was popular, and was supported by all the parties (Hungarians, Entente etc.), so he signed the peace treaty and became the Governor.
Why governor? He could have changed his title, but the last Habsburg king (who tried to restore Habsburg rule) appointed himself Governor not so long ago, and Horthy simple kept the title.