what's this on my roof?

[geoff]

Not much you can do while it's wet, you'll just have to wait till it's hot and everything has dried out
I, too bought a cheap shed but during the first summer I gave it a thorough titivation--
Removed the felt roofing and painted the roof with creosote.
Painted the whole shed all over especially hard to get parts like the eaves and the bottom edges
Reinforced all the joins with extra screws and nails. Re - nailed down the felt with extra nails. Patched a few doubtful parts of the roof felt with spare felt scrounged off my neighbours

This was done about 3 years ago, during a few hot days, and since then it has been completely weatherpfoof. Yes it's a bit fiddly and messy, but you don't have to do it very often. The cost of any materials is pretty low compared to the hassle in having a leaky or rotten shed
The white patches on the inside of the roof are almost certainly fungal, so paint it with anti fungal solution. If rain is seeping through the felt, you'll have to get new felt; I don't think it's that expensive. Your fellow plotholders should be able to help you put it on

[womble]

I've used Onduline on my huge new home built wood shed and it's brilliant, easy to use and lasts really well.
Lets the wood underneath dry out as well, so it lasts longer. Although on my shed I only have supporting beams, not a complete roof.

[Aberdeenplotter]

Roofing felt has a semi-reflective finish to make it less vulnerable to all the effects of sunshine (when we get any). There is also the nail-control thing as mentioned by Aberdeenplotter.
Replacing the sterling board with the plastic sheets would have its advantages, but they DO get brittle after a few years. There is always Onduline, it works well over sterling board (which is what the structure looks like).

I put onduline on my shed at home over sterlingboard. Had forgotten about that. Brilliant stuff. Easy to cut with a hand saw but do have plenty of white spirit on hand to clean it afterwards.