Lawn Mower for winter

I have a Hayter 56 with a roller and whilst it’s brilliant in the summer we get a lot of worm casts in the winter and it’s totally useless as one length of the lawn and the roller completely clogs up with mud, I’m sure it can’t be doing it any goof either as it’s constantly trying to drive with heavy resistance.

My choices are either sell it and get one with wheels instead or buy a cheap mower for either side of the summer that works reasonably well, I try and keep the lawn fairly short and use it to pick up leaves in the winter as we have 2 dogs and it’s really hard to find their poop if the grass is long or covered in leaves.

If anybody’s got any recommendations for something relatively “cheap’ish” that ,may be OK would be much appreciated.