Great question. Let's break it down into parts.

First of all, understand that the Black Death of the 1300's was not one single epidemic, but a series of waves which, depending on the area, began between 1348 and 1350, and then kept reappearing and remitting in echoes until the 20's and 30's of the 1400's.

Second, in the Middle Ages a great deal of the land was not owned by the household who worked it, but by the lord of the region. The title holder reaps the benefits and allows the commoner an amount of the crops. This will become important later.

1- Vacant properties due to the Black Death. Since many households died (in some places, like the area I studied, the realms of the Crown of Aragon, up to 50% of the population), a great amount of property was left unattended. During the 1450's and until the 1480's, there was upheaval because many small rural landowners (commoners, not nobles) wanted to also work nearby lands which were unattended. Now, mind you, the commoner could not just come and take the land, because it had an owner, the lord. We called these areas "masos rònecs", translating to "deserted households".

2- Property and ownership laws vary enormously from area to area, but as a general rule of thumb, a huge amount of the land was owned (not worked) by an absentee lord. This lord would either be a descendant of the warrior nobles of the 1200's (but in the 1400's this lifestyle was mostly on its way out, with the noble being less of a rural warlord and more of an urban politician and courtier), or maybe an ecclesiastic big shot, bishop, abbot, the deacon of a particularly rich urban congregation, or sometimes a rich burgher who had bought land and probably also a title (a common way for merchant fortunes to transition into the landed gentry).

3- The people who worked the land would usually fall into two categories, the serfs (who had many obligations and not a lot of rights) and the free owners (who had no obligations towards any lords but the King). Many of the abuses the lords could inflict on the serfs had been disappearing during the 1300's and in the 1400's they would mostly be declared illegal (thanks to the King's support of the commoners against the nobility), but in the end it all came down to taxes: a serf had to pay taxes for many things, not just crops (using the well, picking acorn and logs, having his house burnt down during a fire, marrying, leaving stuff to their heirs after death... cleptocracy at its finest), while a free man had to pay nothing except for royal taxes and church tithes. This made it so the free farmers would become almost a powerhouse on their own, and after the crisis of the 1348-1400's, they'd be able to gobble a big amount of land by buying it.

4- The Black Death caused such a massive demographic crisis that it made it gave a measure of power to the common folk: less hands to work, less heads to go about. Suddenly the peasants, in a rare moment of class awareness, realised that the lords needed them more than they needed the lords. I don't know how this evolved in other places, but in Aragon this developed into a straight up war between the landed nobility and the commoners (and some nobles supporting them, and sometimes the king, who would use commoner support to undermine the nobles).

5- Many a lord went bankrupt because of all that. Their lands became deserted, they could not attract new farmers, and they raised taxes on the few workers they had left (instead of lowering them, because, after all, having lavish feasts and burnished new armour and keeping the horses fed is expensive, and very essential to the lifestyle of a Medieval nobleman). Some nobles sold titles to rich burghers, others sold lands to the aforementioned rich free farmers. Some took loans (which was not as frowned upon as it had been, but it still wasn't widely accepted; after all, the ideal of chivalric virtue was being generous and using the money to throw luxurious parties; saving money was the opposite of that).

Now, you did ask about the city, but remember that most people lived in the countryside, and it's there where the effects of 40-50% people dying was felt the most.

Most cities had a mass of partially unemployed, or multitasking, population, who were grouped into corporations. If you needed a large amount of people (usually for carrying things from point A to point B, say, blocks of rock for a cathedral, or logs for shipbuilding, or people to dig ditches, etc), you'd go to one of these corporations (more like unions or syndicates, but they call them corporations). I imagine that if you needed to clear the houses of dead people and clean them up, the City Hall would hire these unions... but given how little people would have remained, I'd say either they paid top dollar for them, or whoever came and bought the house would have to hire help to clean it up. Medieval authorities didn't necessarily know many basic things we now know since we're 5 (like washing hands as a way to keep infections or contagion at bay).

As for the "process" the authorities would undertake to combat an epidemic of this magnitude and take care of what's left in its wake, I'm not aware of any concrete after-the-fact policy (aside from isolation during an outbreak. which worked wonders in Milan, for instance). Most big cities in Europe had their own mini republics in place. Places like Paris, Milan, Toulouse, Palermo, Barcelona, Toledo, Nuremberg, Frankfurt, etc, were essentially city-states with an overlord (usually a king), and their ruling class were usually more detached from the common people than we imagine today. Many of these city big shots went to wear the plague out in luxurious country villae (like Boccaccio told in his Decameron).

I'm sure someone with more time and insight than me can scour the sources and give examples. That's all I have!