Natasha yes you can take honeysuckle cuttings in the usual way, or pin down a shoot and cut off and dig up once rooted. They root very easily. I pin down pieces of as many established plants as possible, then you protect your investment against bad winters and plants can get old and woody and a replacement in a pot saves you money. When the cold frame is too full, I exchange plants with others or give them as presents. Whatever, if they don't root you have only spent time and a pot of compost, worth the risk. Give it a go and enjoy the outcome.