my plot is also at the bottom of a slope so in very wet seasons can get a little flooded - this is the first time it's got bad again since 2007. That's good advice about digging in some sand to help drainage and most important, wherever clay soil is compacted water will always pool there so avoid walking on the bed.
At the end of the day though, if the plot is at the bottom of a hill it will always have potential for flooding as water will drain there in very bad weather. Digging a ditch to funnel water away from the bed sounds like a good option but may I also suggest creating raised beds too. This has worked well for me - most beds are only about 6-8 inches high and the key is not to walk on them and every year keep adding compost, manure and some sharp sand - ideally I'd like them a little higher but as you have also noticed, it costs money so work on improving the beds every year and not expect to do it all in one go.
The good news is ... in drought conditions, your beds won't dry out quite so quickly as those plots on upper slopes
Would covering it with tarp help the soil to dry out more quickly?
No. It will just keep the water under there longer and prevent it from evaporating as we have established that it wont drain away. Cover it before it gets wet, uncover it when the weather's dry. If you can keep turning over the soil into big lumps it will dry out a little bit faster.
I'm inclined to agree that putting a cover over when its wet is not such a good idea, but if water is draining down from upper areas (gravity), the water will still pool even under the cover and as I found out for myself on a bed I'd left under cover (for weed suppressant), the clay soil seems to have got even worse when not exposed to weather. The best option is to leave uncovered.