We make the link between “last” and “minute” easier by making the t silent. The t at the end of “minute” is pronounced right down in your throat where you make a /h/ sound. We’ve practised this before. This makes it easier to get to the /ʃ/ sound.
This is a noun phrase, used, for example, in a phrase like, “I have to do some last minute shopping.” That means shopping when there’s not much time left. Don’t panic though. Christmas, if you celebrate it, isn’t till Sunday. Plenty of time..