DC12V 0.45A mini blower?

Use could use either one. For the 6800 MA converts to 6.8 Amps. The fan only requires 0.45 Amps, more is okay less may burnout or short the power source.

Welcome to Fluther !

Please check the specification again. Are you sure it said 3800ma and 6800ma? Or did it say 3800mAh (milli Amp hours). Those are typical energy storage numbers.

12 volts x .45A = 5.4 Watts. You want the source to be capable of delivering 2x that, or 10 Watts so it can cover inrush current and start up loads. Odds are good the power supplies/batteries you found will work They should deliver 1 amp easily but check the specs.
The 3800 mAh unit will (theoretically) run your fan for 8.4 hours -. figure 6 hours
The 6800 mAh unit will (theoretically) run it for 15.1 hours – figure 12 hours.

@LuckyGuy is correct be sure you check. On power supplies power is usually described with a voltage and a current ex: 12v @ .5A max. Do not exceed this current or even come close to it, double what you need is a good rule of thumb. Batteries will display power as mA/hr because there is a finite capacity. It makes calcs on “how long a battery will last under a certain load” easier. If you are using a powersupply you are fine, if you are using a battery you are probably fine but you would need to give us more information on exactly what power supply/battery you are using.

The model number is HAJ 12680 an the output says
12.6–10.8vdc 6800mAh.

Mah is a unit for measuring how much capacity a battery has, not how much power something uses.

@Neverless As expected.
My numbers are correct . The HAJ is a super polymer Li-ion. Plenty of current capability. It will run your fan for 12 hours without recharging.

The important thing is the voltage. If it requires 12 volts and you are supplying 12 volts, you’re OK. The unit draws amps (through the fan’s resistance) which isn’t very much, so just about any size wires to the unit should suffice. You can’t make the unit take more amps than it was designed to handle.