Moss + dogs = lawn disaster

Hi all,
I have live in a flat in a very soggy part of Scotland and we have a shared lawn out our back. The garden is a reasonably nice space to be thanks to a neighbour who has since moved out and taken her green fingers with her but unfortunately the lawn is a disaster.
We have two dogs and there are another two that live in the building. The grass was mostly moss (I suspect due to the fact parts of the garden are very shaded and a lot of water can gather on the grass making it permenantly sodden) and when the dogs have been running on it they have just ripped huge chunks of it up leaving bare patches which have now turned to mud. It honestly looks awful and I'm sure our neighbours who don't have dogs must be a bit annoyed at us for destroying it over winter.
I'm willing to do the work it takes to fix it but I know NOTHING about lawns or gardening in general. Can anybody suggest anything? We haven't got a lot of money to spend and the lawn doesn't need to be perfect - just now the muddy, soggy mess it is just now!
Any suggestions would be great for this gardening newbie!

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