If I already have my UK Customs Clearance, am I ok to ship to/from Europe?

No, you will also need to make sure that the EU customs clearance is arranged, as you require an export and an import for each movement. 

If your buyer/supplier in the EU is arranging this, all you need to do is input the EU MRN (Movement Reference Number) into CustomsLink alongside your UK customs entry and the Haulier will be able to complete the movement.