The right of appointing the local vicar, correctly called the advowson, was granted to corporations (this is not strictly true it usually had to be purchased) - also previously mentioned, but it does not give you any real ideas of the importance of this right.
The parish vicar was a very important person, and had the right to make further appointments to various committees and town bodies, including the oversight of the poor funds, and also had a part to play in appointing the town council.
Usually the vicar would be able to appoint around one third of the council from local clergy- the aldermen etc, and this was a crucial position, as it often meant the vicar could hold the balance of power in many town councils between the merchants and the landowners.
You could not get yourself appointed to the living as vicar unless you had certain views, but equally the local vicar could only be removed, if at all, by senior church officials. Once you had appointed the vicar, that was it, you were stuck with him, so it had to be right first time.
Put plainly, everyone wanted a vicar who would do what they were told, but since it was such a powerful position, it was better to arrange things so that this person was part of a local wealthy family - keep all the power in the hands of the controllers - whoever they were.
The advowson was generally owned by absentee senior clergy, who often were not all that interested in the appointment, acting on the recommendation of the local squire or whoever held the manorial rolls, but in towns and cities, such a position of power had commercial value.
The advowson would be purchased at very great expense by the local councils when they became incorporated, hence the term corporation to mean town council.
For the corporation, it meant that they could effectively control all appointmemts to the various town officials positions, remember there was no such thing as a local election so the faces only change when someone dies and a new appointment is made by the self serving corporation.
Some City Corporations did not have any other access to national power in terms of members of parliament, but they often wanted more regional power, to control trading etc. Having a total control of the local Corporation meant you could pass local by-laws which would do things such as set apprenticeship rules, control the quality of goods, decide who would be allowed to become merchants and many other things.
This total control made immense differances, at one time the two major trading wool and cloth towns in England were Leeds and Wakefield - with Wakefield being the foremost. (and this represented around one third of all the monetry value of Englands exports) These two towns are no more than 30 miles apart and make an interesting study in Civic politics.
That total control of the Corporation in Leeds allowed it to undertake activities such as canal building - which required a huge amount of money and ultimately meant that Leeds forged ahead to become Englands fifth largest and most prosperous city, and for Wakefield to fall back into a middling sized city with much less regional importance in terms of trade, finance, just about anything you can name.
By the time you speak of, the advowson was still important, however nothing like as influential as perhaps during the Georgian period, in some country estates, the local church was still entitled to a tithe from all the farms in the area, which meant a reasonable income.
In Leeds during the 1920’s and 1930’s the local vicar was influential enough to have large chunks of the city demolished due to unhealthy housing and other notorious slum areas.
There is much more of just how the advowson and the local vicar had the ability to wield power and huge influence locally, in a little country estate he would be one of the most imprtant people in the area, and a local squire would have had to have extremely good connections to be able remove him at will, most likely in the House of Lords and in the senior church hierarchy.