As electric motors or machines get larger in size then the current that they draw (especially on start-up) can exceed the 13A that 13A plugs are fused with and so the fuse blows. There is a grey area - depending on the size and loading on the machine -where sometimes it might blow..sometimes the motor might start up OK. I'll give you an example. My spindle moulder can be fitted with anything from a light aluminium block to a huge lump of steel. The starting current needed to get the motor turning with that lump of steel will blow a 13A fuse for certain. Which is why those machines have 16A plugs on them.
The 13A plug in your adapter (with the 16A socket ..you say plug but I think you mean a socket) might simply be just that...an adapter to let the owner plug their machine into a 13Amp socket. Chances are though that whoever made the adapter up, got a bit of wire and wrapped in around the terminals where the fuse should go. Not a good idea.
Ideally you'd run a dedicated cable directly from your consumer unit to a 16A socket. Called a radial cable. However, the sizing of the cable depends on factors such as length of cable etc.
Get an electrician in.