Widen the bottom of the gap with a chisel and hammer. Place the tip of the chisel at an angle against the inner wall of the gap and hit the flat end with a hammer. Chip down the length of the gap.
Scoop chunks and debris from inside of the gap with a wire brush.
Measure the surface width and depth of the gap with a measuring stick. If it is deeper than 1/2 inch, fill the bottom until it is 1/2 inch from the sidewalk surface.
Add a light coat of bonding adhesive inside and around the surface of the gap with a paint brush. The adhesive helps strengthen the bond between the fill and the sidewalk.
Stir a container of vinyl-reinforced concrete patching compound with a wooden stirrer. If the surface of the gap is less than 1/2 inch wide, use a putty knife to fill the remaining 1/2 inch expanse of the gap.
Smooth the compound around the surface of the gap with a trowel to make the filled gap even with the sidewalk.
Stir pre-made sand mix concrete with concrete fortifier if the surface of the gap is wider than 1/2 inch. Fill the remaining 1/2-inch gap with the fortified sand concrete instead of the vinyl patching compound. Smooth the concrete over the sidewalk so the surface is even.
Cover the patched gap with moist burlap to dry it slowly. The added moisture prevents the fill from absorbing moisture from the old concrete.