Taking a photograph in ambient light typically means changing the cameras setting's to attain the correct exposure. Taking a photograph with studio lighting allows you to set desired camera settings and adjusting the lights to match.
Mixing the two means mixing the two ways of attaining the correct exposure.
We already know that shutter speed does not affect exposure in flash photography due to the extremely quick burst of light, leaving aperture and ISO to affect exposure. We also know that ambient photography is affected by the entire exposure triangle.
If we set our ISO to it's lowest to gain the best quality file output, this leaves aperture to control flash exposure leaving shutter speed to control ambient exposure.
For these images I first metered the ambient light in camera, I wanted a large depth of field to keep the sky in focus so I set an aperture of f/16.
At ISO100 this gave me a shutter speed of 1/50 to expose correctly for the sky. However, I was hand holding the camera I wanted a a faster shutter speed to ensure a sharp image. Changing the ISO to 250 allowed me a shutter speed of 1/125.
I then set up a studio light pointing at Katie and metered the light, making sure to set the ISO value on the light meter to the same as my camera, and adjusted the light until it read f/16 as the correct exposure.
ISO250 f/16 1/125
ISO250 f/16 1/125
Here you can see how lower aperture affects both ambient and flash exposure. If I wanted to affect the flash lighting only, I would also have to compensate with a higher shutter speed. As I opened the aperture by 2 stops I would have to increase the shutter speed 2 stop to 1/500.
However, altering the shutter speed alone would only affect the ambient light.
ISO160 f/8 1/20
ISO160 f/8 1/125
Here all camera settings are kept the same except the shutter speed. You can see on the left at 1/20 Katie and the background are at the same exposure creating a flat looking dull image.
At 1/20 you could normally expect movement blur from camera shake however the flash helps to freeze motion, similar to my studio splash shoot.
On the image to the right, with 2 and 2/3 stops difference in the shutter speed, you can see a dramatic difference in the background due to the ambient lighting being underexposed. Katie is also a little darker than the left image as she was also being lit by some ambient light.
Problems can occur with ambient and flash when equipment limits desired camera settings.
For example, if I wanted to shoot Katie at f1.4 to create a really shallow depth of field, even at ISO100, particularly in daylight would require a shutter speed quicker than 1/200 (maximum flash sync speed).
To remedy this you could use HSS and reach a higher shutter speed without worry about shutter sync issues or opt for a neutral density filter.
ND filters screw onto the front of a lens and act a little like sunglasses. They block out the light entering the camera by a determined amount of stops allowing a really birhgt scene to be shot at wide apertures and low shutter speeds.