Water logged garden and lawn

Afternoon, I need some advice for the back garden of the house I'm renting.
At the bottom right corner of the back garden we have a flower bed, and around the garden we have a gravel border, with a small path in front of the corner plot. My problem is the right hand side 1/3 of the garden and lawn never gets any sun. Its caused moss to grow in the lawn and over the small path thats on that side. The corner bed is waterlogged, nothing grows in it, and there is now standing water on the path along the front of that bed.
Can you give me some ideas on how to rectify or improve this, not so much the moss but the water as thats the cause. Unfortunatley it can't be as drastic as getting rid of the bed , as it's a rented property. Also while im on, whats the best way to bring my front lawn back to its best, its got tufty and has developed bald patches from the snow falling before i could clear the last leaves from it.
Would be grateful for any advice, as I'm eager but a total novice ! Also do'nt have alot to spend. Either replies on here or direct email is welcome

Cheers
James