Robins' nesting in my extremely messy shed

Robins are nesting in my shed in my garden tool cupboard, which is low-level.  My shed is super messy, basically a dumping ground for old boxes, tools, paint equipment, bed frames etc .. and im really nervous the fledglings risk getting trapped in amongst all this stuff.  I cant open the shed door as the neighbours cat is constantly keeping guard on the roof of the shed.

I was wondering if it would be worth disturbing the birds and trying to throw tarpaulin over as much of my stuff as possible. so that if they land anywhere, hopefully they would just slide to the floor.  

Very much looking forward to receiving any guidance.

kindest regards

Louise