The invention relates to a portable, hand-held work apparatus such as a chain saw, a cut-off machine, a brush cutter or the like with an air cooled combustion engine. A cooling air blower (4) which includes a cooling air spiral (6) and a fan wheel (5), moves a cooling air flow (13) for cooling the combustion engine. Further a combustion air channel (30) leads from the cooling air blower (4) to the combustion engine (2) and branches off from an air output window (20) provided in the base (14) of the cooling air spiral (6). The diverting device (21) includes a guide wall (22) which extends into the cooling air spiral (6) between the fan wheel (5) and the air output window (20). In order to divert a large volume of combustion air (19) with minimal disruption to the cooling air flow, the pass-through cross-section of the air output window (20) tapers in the direction of the diverted combustion air (19) from a first pass-through area (29) to a second pass-through area (31).