Network Rail is basically a wholly government owned company with an independent board of voluntary shareholders. NR owns all stations except those built by entities other than it, for example GMPTE in Manchester built Lostock Parkway and Northern operate it on a management contract.
Theres multiple types of franchise agreement and a couple of rounds of franchise award so no franchise-government relationship is identical, it is also ever changing so you will have a book that gives you a snapshot but it will never be current. Fundamentally they broke down into three types though:
'No Growth' Franchises like Northern Rail, when it was franchises the government didnt think passenger numbers would rise so it was thought little investment was required, a contract was therefore set up to maintain the status quo. Firstly revenue projections were compiled for each route then quarterly the companies revenue is independently audited, any surplus the government takes straight away, any shortfall the government pays out at the end of the financial year. Because its done on a per route basis the company could in reality be paying premiums to the government for some routes at the same time as recieving subsidy for other.
Over 106% predicted revenue the government takes 80% cut
100-106% predicted revenue the government takes 50% cut
94-100% predicted revenue the government gives 50% of the shortfall
under 94% predicted revenue the government gives 80% of the shortfall
The second type of franchises are 'Premium' franchises, those expecting large growth or which need a lot of early investment which will pay off in later years, there are some aspects of revenue support for example if revenue was considerably below the governments projections they would recieve revenue support. However rather than looking directly at revenue these franchises have a graduated system from annual subsidy to annual premiums which is supposed to reflect the companies making efficency savings and gaining rewards from investment. For example in the first year the company recieves 400m, 300m in second, 200m, 100m, then the company starts paying ever larger premiums to the government 100m 200m 300m 400m etc... They are expected over the lifetime of the franchise to actually bring a net profit to the government.
The third type are 'Management' type franchises, these are franchises expected to be prone to disruption from engineering works, have unstable passenger numbers, etc.. In these the government takes all the revenue and all the shortfalls paying the franchisee a fixed amount to manage and operate the franchise usually with some performance bonuses. The franchisee is able to propose investment over and above what was specified and if approved it will finance the work itself and make the money back either from agreed higher management payments or through extensions to the length of the franchise.