From 1227 to 1234 we are sort of in betweeners - the minority has ended, but Henry's government in still dominated by the old guard, people like Hubert de Burgh. But it doesn't go well - money is still tight, Henry's campaigns in France aren't great, and in 1232 Peter des Roches, the old enemy, is back in the country.
The period between 1227 and 1258 is tagged as Henry's personal rule. The first period within this might be considered to be 1227 to 1234, from when he announced at a magnum consilium at Oxford that he would take control of the royal seal; until 1234, when the fall of Peter des Roches signals the end of the system of Justiciars or high profile leading men.
The fall of Hubert de Burgh
When Henry ends his minority, Hubert at last cashes in - he is made Earl of Kent, is assigned 2 honours and the welsh border castles of Grosmont, Skenfrith, the White Castle and others.
But by 1232 his government and reputation is in trouble; the barons didn't like such a low born bloke having such riches. Hubert's power had begun to rub the church up the wrong way, and Henry managed to blame him for the failure of the French campaigns and the lack of available cash. So on the return of Peter des Roches, Hubert is in trouble - and in 1232 he is imprisoned in Devizes castle, and des Roches' man, Peter De Rivaux takes over.
The Fall of Peter des Roches
Peter des Roches was an even worse choice. At least Hubert basically believed in Magna Carta and consensual government between king and barons - Peter rode roughshod over all of that. Plus he completed failed to improve the royal finances in anyway, and by 1234 the Earl of Pembroke was in revolt.
It took Edmund of Abingdon, the new Archbishop of Canterbury, to bring peace. He made Henry accept that he made the wrong choice - and Peter de Rivaux and Peter des Roches were removed. From here on in Henry aimed to rule without he great officers of state, though strictly speaking Stephen Seagrave was Justiciar for a short period.