A109E Captain reported ATC failure to provide separation resulted in evasive action taken to avoid an NMAC.

Narrative

My crew and I were on the ramp in front of our company hangar; an area that adjoins ZZZ via Taxiway 1; on the approach end of Runway XX at ZZZ. We were coordinating for departure VFR to the Location X area; a roughly 340° heading off of ZZZ at 2000 to go pick-up a patient. Tower cleared us for departure from our area and announced Cessna traffic originally on a left base for XX. Just prior to our takeoff and during subsequent contact with Tower; the Cessna was announced on short final; but our takeoff clearance was never changed or rescinded. Upon taking off on approximately a 040° heading to expedite runway crossing; while scanning for the traffic; my Flight Nurse sitting in the aft right of the aircraft announced that the traffic was very close to us at our five o'clock and called to break right; which I did. During that maneuver; the TA alarmed us to the traffic as well. The Cessna passed safely behind us; but the distance was far too near for comfort. In retrospect; the most conservative response to the Tower's announcement of traffic would have been to simply remain either on the ground or in OGE (Out of Ground Effect) hover over our ramp until the Cessna was on the runway and in sight. Further impacting the situation is the fact that ZZZ's Runway is currently undergoing construction and has been shortened by about 4000ft; bringing the XX Base leg far closer to our hangar than it normally would be; and making traffic on that leg relatively higher to where we are all accustomed to scanning. During subsequent debrief of the event amongst the crew; the aforementioned practices were suggested for all future flights; and the event underscores the importance of maintaining vigilance to all ATC instructions and clearances to verify that they do not contravene Crew and aircraft safety.