That "vertical 'vent' hose" is exactly that. It's called an Air Admittance Valve (AAV)*.
The purpose of the AAV is to allow the line to drain properly when a regular vent line cannot be installed through the roof. The drawback to AAVs is that they fail. It seems that yours has. The suggested course of action would be to replace the AAV.
I believe that B&Q is your equivalent to one of our "big-box" home improvement chains like Lowes or Home Depot. They will probably have an AAV that will work as a replacement. Yours appears to be permanently connected to that bit of flexible pipe, so unscrew the lock ring at the bottom of that pipe and take the whole thing to the store with you. The goal is to find one whose opening is the same diameter as the one you brought with you and isn't much taller, since you don't seem to have much additional head room under the cabinet. I can only imagine that your screw fittings will be standardized as ours are in the US, so the threading on the new one should match up with the threads on the old one.
When you get home, simply screw the new one on. If you have a bit of leakage around the threads while the washer is draining, screw it a bit tighter. If you have to get out a wrench, do not turn it with a wrench more than about 1/8 of a turn! These fittings are designed to seal when hand tightened, so anything more than that won't help and will only cause additional damage. If you can't get it to seal, ask a new question about how to stop that leak.
Additional note: in the US, at least, there are very specific requirements for where an AAV can be installed and how it must be installed. I'd presume the same is true in the UK. I'd recommend checking your local regulations just to be sure your installation was, and remains, code compliant.
*This was the first link I found that seemed like it was just an explanation and wasn't really trying to sell anything. No relationship with the linked page.
