How to correctly use an Avenir Airsource One bicycle pump
That appears to be a standard flip-up chuck, and it claims to be "universal".  If using on a Schrader valve you just push it on and flip up the lever.  If using on a Presta valve you unscrew the knob on the Presta valve first, then push on and flip up the lever.

With some of these you need to not push the chuck quite all the way on if using on Schrader.

On the other hand, if the tube is fully deflated, you may need to press the tire with your thumb to hold the valve in place (keep it from sliding back through the rim) as you press the chuck on, so that you can get the chuck on far enough to hold.  (My suspicion is that this is your problem -- the valve is sliding into the rim so the chuck does not get on far enough.)

If need be, you can sometimes tighten the nut around the chuck opening to achieve a tighter fit (or loosen if the fit seems too tight).