Replacing halogen under-cabinet lights - connection question

I’m replacing some old halogen under-cabinet lights but am not sure how to connect the new LED replacements.

The old halogen lights had the following connector which is not compatible with the new lights:

The new connection looks like this, and in turn connects to an LED power supply which then connects to the new lights:

My question is, can I just cut the plug off the new cable and wire it into the circular connection box, the contents of which looks like this:

I realise doing this will remove the fuse from the circuit, but is that an issue? Is it safe to assume that the connection into the connection box above is fused in some way?