Wood Pigeons bringing sticks to my balcony

Wood Pigeons- I have a feeder on a juliet balcony and the Blue tits and Robins have been visiting. It has also got the attention of Wood Pigeons and I let them feed but they have become a nuisance crashing into small birds, trampling on my flowers and wolfing down the food. Recently I have been walking near the window when they arrive to deter them as the small birds don't get a chance. I wanted to let them have some food as they are calm birds, but they are quite problematic and draw attention to the feeder which I am not sure is allowed.

This morning one brought a stick up and then shortly after another stick. Is it trying to nest in my hanging basket or just bringing a peace offering in exchange for food?

I live in an apartment block and I think the nest would get me in trouble and I'd be asked to remove the feeder.

Should I remove the sticks or see what happens? I was considering removing this particular hanging basket as it has become a landing zone for the Pigeons.