Actuallly, weight does have an impact. The three main loads on any car (EV or ICE) are rolling resistance, air friction, and accessory load (hotel load). Rolling resistance is proportional to weight and the type of wheel (ie tyre). air friction is proportional to the drag coefficient, the frontal area, and the density of air ('which changes with temperature and pressure). Accessory load is everything the car is using that doesn't propel it, eg. air conditioning, battery cooling/heating, power electronics etc.
At slow (city) speeds, air friction is minimal, as it depends on v squared (speed * speed). Double your speed = 4 times the drag. Rolling resistance doesn't depend on speed. You also need to haul the weight of the car around, going up hills uses energy proportional to the weight of the car. Going down regens a percentage of the energy used going uphill, but not all of it. At freeway speeds, air friction is the dominant loss as it quickly increases with speed.
As well as the physical losses, there is also the efficiency of the drive train. You may need x kW of power to travel at y km/h, but if the motor/electronics is only 50% efficient, you need to supply 2 * x to deliver x at the wheels. It means efficiency is quite complicated, but to boil it down:
(assuming the rest of the car is kept constant)
* a light weight car will be more efficient than a heavy one
* a smaller (area) car will be more efficient than a larger one
* a sleeker car will be more efficient than a less sleek one
* better electronics is more efficient than not so good electronics
So a small, light, sleek car with good electronic efficiency is the holy grail - check out the Aptera. Appologies for the size of the post - it grew from what I intended