The four men in charge of Prague in 1440 were Menhart of Hradec, Hanuš of Kolovraty, Pešík of Kunwald and Pavel Dětřichovic.

Prague was not yet a single entity at that time, with its constituent towns being basically independent of each other, although there were attempts at joining together the various parts well before the lasting unification of 1784. In fact, just a few years after the founding of the New Town, Charles IV ordered the demolition of walls between the Old and New Towns in 1367 with the aim of joining the two. However, the Old Town still retained its privileges and many of the less desirable elements were forced into the New Town, and the resulting local unrest forced the emperor to revoke his decision just a decade later, re-establishing the division.

Another attempt at unifying the towns came during the turmoil of the Hussite wars, when under the leadership of radical preacher Jan Želivský the two towns were joined together for a short time during the 1420s. But, just like the Hussite movement itself, this union was soon fractured, the Old Town siding with the conservative Utraquists, while the radical New Town continued to support the Orphan faction. The enmity culminated in 1434 with Old Town forces successfully attacking the New Town, subjugating it for a time and strengthening the position of the Prague Old Town as the foremost city of the kingdom.

Despite the above, the future disctricts of Prague – the Old and New Towns, the Lesser Town and Hradčany – were separate municipalities with their own local governments. Alongside various positions in the city bureaucracy such as magistrates, scribes or tax collectors, some of whom were appointed by the king, while others would be voted on by the burgeoning bourgeoise, the towns of Prague were governed by city councils. To further complicate matters, the burgrave of Prague Castle exercised a great deal of influence over the city as well.

Nominally, these councils would be subject to and appointed by either directly the king, such as those of the Old Town, or an intermediary such as the king's chamberlain, as was the case with the smaller and less powerful Lesser Town. Additionally, the councilmen would usually appoint a chief official from their ranks – "purkmistr", from German Bürgermeister, meaning simply "mayor" – voting on a new one every four weeks to prevent abuses of power.

These various links and hierarchies would change and evolve fairly rapidly over time, but the privileged position of the Old Town, and conversely the somewhat disadvantaged position of the other quarters would be something of a constant. As such, this also helped fuel the previously mentioned enmity during the Hussite wars, which, incidentally, were sparked by the first Prague defenestration in 1419, when a mob stormed the New Town Hall and threw several councilmen out of a window. Obviously, the conflict also saw changes in local governance, and Hussite loyalists taking over positions of power and establishing their own councils.

Several of these changes would actually survive the Hussite wars, with Sigismund confirming many of the cities' rights such as naming their own magistrates, strengthening their own judicial powers, or non-Utraquist foreigners not being able to hold office. The emperor also reinstated regular city councils, with the Old Town council numbering eighteen councilmen and the New and Lesser Towns having twelve, but the cities would become ever more independent.

Attempts in the following years to curtail this growing power of Prague and cities in general were unsuccessful, anfor example when Albert II of Germany tried to force his own councilman into the Old Town Hall and in retaliation the council had the man executed. The interregnum also did little to lessen the independence of the cities, and city councils went largely unchecked, as councilmen were not being replaced and enjoyed a great deal of power and autonomy with little, if any, oversight.

It should come as no surprise that the cities opposed the attempts of Bohemian nobility to appoint a magistrate who would enforce the now often overlooked royal rights, but they suffered a setback in 1448 after George of Poděbrady attacked Prague to suppress Catholic opposition. The city councils were restored to a semblance of the former order, but ultimately the damage was done, and in the coming decades the cities would keep pushing for more rights and privileges.

This, finally, brings us to the answer to your question. In addition to all of the above, even before the death of Albert II and the interregnum, the situation in Bohemia was uneasy at best. A sizeable part of the nobility opposed the new king and plotted to take control of Prague in order to prevent his coronation. However, before they could act, the moderate, pro-Catholic and pro-Habsburg faction led by the powerful nobles Oldřich of Rožmberk, Hanuš of Kolovraty and Menhart of Hradec struck first, ousting or outright arresting Prague officials and nobles involved in the scheme in early 1438.

Matters did not get any less complicated after Albert's death in the following year. Landfrieden were established throughout the kingdom – regional councils passing and enforcing special laws to uphold the peace and order – and six nobles were appointed as caretakers to administer the land. Three of these were Oldřich, Hanuš and Menhart, and in 1440, Prague was firmly under the control of the latter two.

Hanuš was a Sigismund loyalist who initially opposed the Hussites, only supporting the moderates and switching sides after his holdings had been besieged later in the war. After the war he oversaw the Prague Landfried together with Menhart, and became the hetman of the three Prague towns. After George of Poděbrady attacked Prague in 1448, he fled and died two years later at one of his castles, which George would later take over and grant to his sons.

Menhart, originally a Catholic, became one of the leaders of the moderate Utraquist faction during the Hussite wars while at the same time trying to maintain a good relationship with Sigismund, and later returned to the Catholic faith. In 1437, Sigismund appointed him the burgrave of Prague, one of the highest – and, during the interregnum, quite possibly the highest – offices of the land. He was not a popular figure, and following the fall of Prague to George of Poděbrady, was imprisoned and died, with unproven rumours of poisoning surrounding his death.

His position and both political and military power as the lord of Prague Castle was formidable on its own, but he also had control over the city proper. Prague was ruled by his supporters, Pešík of Kunwald, the Old Town bürgermeister, and Pavel Dětřichovic, the mayor of New Town. Incidentally, these two have not left these offices after the customary four weeks – illustrating well the aforementioned political climate and the cities' unchecked independence of the time – staying in power for several years under Menhart's protection. They avoided their master's fate, but, like Hanuš, also fled the capital after 1448.

So as you can see, we know a pretty good deal of what the local governments of Prague would have looked and worked like in the 15th century, as well as who was in charge in 1440 specifically. On the municipal level, it was the mayors Pešík and Dětřichovic at the heads of the city councils, and these were in turn controlled by the burgrave Menhart, who alongside hetman Hanuš also exercised a great deal of power over Prague as well as at the highest level of contemporary politics of the Czech lands.