I believe it's a Euphorbia, because of the flowers. I can't see any reason why you couldn't re site it.

When moving just be aware of the fact some folk get a reaction to the latex in the stems etc. so I'd probably wear gloves.

Looks like one that would root easily if you took a couple of cuttings. Just let the cut stem scab over before potting.

If all of it is sprawled across the ground, and its not huge (can't tell from the photo, it might be showing the tops of stems 2 feet long) it's probably Euphorbia myrsinites

Euphorbia myrsinites was my first thought too.

Thank you very much for your replies, they are very helpful.
Moon grower yes I became aware of the sap when I accidently broke a piece off. I have to be extremely careful regarding reactions to certain plants. I have had an extremely bad reaction to Primula obconica so I'm very careful but sometimes you forget .?