Early on he describes a painting of Christ in Majesty with the Church and Synagogue. Near the end he describes Henry III's extortion of money from the Jews, whom he accused of fraud--to Matthew's satisfaction but modern suspicion.
In the last year or two he describes wild weather, earthquakes, and tsunamis, and he describes the destruction of the 7'th Crusade. (St. Louis was unable to keep the competing egos in check, didn't have good planning or support, and seriously overreached.) Some people "took the cross" in order to collect donations, but without any serious intention of going to fight.
This bit of description has an interesting assumption built into it... Matthew is being a bit snarky.
The above-mentioned special clerk of the lord king, whose wealth attained to episcopal heights ...
An empty church is one without a priest or bishop to receive the income from it, and the king, except when there was a special exemption, had veto power over who got appointed. Unless he decided to ignore the exemption. And Innocent IV never had enough money either,, and any appointments or other requests had to be well-lubricated. Quite a bit of the chronicles describe one or the other kind of novel extortion and waste. Innocent and Frederick were at war, and fortunes of war meant that large amounts of extorted money wound up lost.
He doesn't describe traditional exactions, unfortunately.
This was an era in which the barons fought the crown from time to time, and we, from the perspective of the heirs of the winners, think the dominance of the crown was the right goal--centralize the rule of law. But at the time, it wasn't quite so obvious, and the crown was exceedingly arbitrary--and stupid. Some themes seem quite familiar: in order to maintain a reputation for generosity to the poor, despite his lack of liquidity, Henry III visited London and told Londoners that their traditional New Years' presents would be given to him this year.