A few months back I had a problem with the cold tap in my upstairs bathroom — no water would come out after turning it on. I also noticed that the flexible tap hose underneath the basin was damp and sometimes dripping a little bit. So I got a plumber out who fixed the lack of water problem by replacing the tap thread. I rather assumed that that was also the cause of the flexible tap hose dampness, but I was wrong. Then I noticed that there was a tiny gap where the tap wasn't 100% over the hole, so I loosened the back nut and repositioned it to close off the hole.
Unfortunately that didn't solve the hose dampness/dripping problem. So now I'm thinking that the problem is perhaps due to the flexible hose being past its best. So I'd like to change the hose.
As a DIY/plumbing newbie, I don't recognize the connection between the hose and the plastic piping. At first I thought it was a standard braided hose, but then when I unscrewed the nut, the thread of the hose is connected to what looks a bit like a compression olive. Is anyone familiar with the connection? And would a reasonably handy but inexperienced DIYer be wise to fix it himself, or is this a job for a pro?
EDIT: To assist with Duncan's helpful answer, I'm adding two images: