yes just be carefull not to disturb the root system

Which particular 'poppy' are you referring to Dannyboon?

depends on which one. if its the perennial type such as the large oriental ones you are better to wait until late summer early spring. but if you leave bits of the root behind they will regrow there.

If they are the annual type you'd be better collecting the seed and sowing it where you want them next yr.

I have just moved one that was caught up with a Rose. It wilted at first but it has recovered.

I've only got experience with some very large annual orange ones with glaucous leaves (sorry no idea of the variety) and you can only move them successfully when they are very small.