The best way to repair a 'push button' Barlow winch is throw it into a trash can !!!!!
Yes, you push down on that 'smooth cover-like' object down in the center of the bore and this will release the spring that holds the barrel to the spindle. The problem with such 'push button' Barlow winches is that they do separate under the right conditions and under high load. I had one come apart when applying a heavy load, the barrel separated from the spindle, the barrel traveled along the tailed line, hit me square in the face, knocked me out momentarily (saved by my harness from going overboard), smashed my glasses 'into' my face, blood all over the deck, only took 8 stitches to close an additional 'eyebrow' ..... all because the spring under that so-called push-button Barlow became weakened and could no longer hold the barrel to the spindle when at heavy load. .... the worst part is that this was my PRIMARY mast base winch, a winch that was used many times to help get me up the mast.
After this incident, all my 'push-button' Barlow winches went directly to the trash can; if this winch of yours turns out to be one of the 'push button' release type Barlow's, I strongly urge you to also 'trash it'.