Is there a way to only display markers for the current mission?

I've just got out into the city, and already I have two main missions, and 4 side missions, one of which has 3 different locations which I need to go to.

This adds quite a lot of clutter to the screen, is there any way of cutting it down?

You can choose to disable individual missions. This will remove the mission marker from your HUD, but the mission can still be completed.

Bring up your menu, then go to the missions tab (or just press U). Each mission should have an "INACTIVE" button that appears when you put your mouse over it — clicking that will hide its mission marker. To undo, simply click the button again.

The indicator is in the upper left, the toggle in the lower right corner.