Carton or cover of board sheet material including at least four rectangular lateral walls connected together by parallel join lines and an assembly of upper flaps intended to form an upper face of said cover and connected to said lateral walls by fold lines orthogonal to the join lines, said upper flaps comprising two inner flaps and two outer flaps, each of said two inner flaps being secured to one of said four lateral walls situated between two of said four lateral walls respectively furnished with one of said two outer flaps,
- wherein said carton or said cover included two sinkable corner regions situated respectively on either side of the carton or of the cover, straddling the fold lines corresponding to two opposing walls of said four lateral walls,
- wherein said carton includes an elongate handle comprising a lower band made of sheet material formed of three separated parts, one central part formed by a detachable part belonging to at least one of said outer flaps, and two end parts formed by rectangular parts precut or cut out from an upper peripheral border of said opposing walls and a reinforcing upper band secured to said lower band having ends fixed on each side to said opposing walls, underneath said sinkable corner regions with respect to the upper face of the cover, the handle encompassing or extending opposite said sinkable corner regions so that, when the handle is pulled substantially perpendicularly to the upper face of the cover, handle portions situated opposite said sinkable corner regions sink into the carton, squashing and/or penetrating said sinkable corner regions, and wherein each of said sinkable corner regions comprises a lower part corresponding to one of said end parts partly precut or cut out from one of said opposing walls and extending downwards over a specific distance d from a corresponding fold line and a slit or frangible upper part extending in the upper face of the cover.