When i switch on the engine it sounds that it is going to stall and seems to me that engine power produced isis less

2012 really ? what country? we have no 626 year USA that year
or do you mean the Mazda 6"? 2.5 or 3.7L engine?
cut out means what, be more exact in what that means
ALL DC power and lamps and engine and radio die , on the fly.?
or just engine, all lamps good.
lost power means engine power or DC power or both.
engine stalls. are complex over 50 causes in engine or EFI.and more.
can the stall (stall means engine goes to 0rpm) but can you prevent the stall with a fast right foot?????
does the engine restart easy, at stall? or not.
is the starter motor always working(cranking), after the stall?
stalls may only happen on hot or cold engine or both.
bad spark is #1 cause of stalls. (and lots more ) try new spark plugs.>?
EGR stuck open
IAC stuck closed(idle valve gunked up)
battery died, or alternator dead or intermittent.(or going off line for battery gross shorts or opens)

mazda 6
Mazda since 2002, replacing the long-produced Capella/626.
and
The 626 replaced the 616/618 and RX-2 in 1979 and was sold through 2002, when the new Mazda6 took over as Mazda's large family car. 4,345,279 of the 626 and Ford Telstar models were sold worldwide.

post real car model names and real engine size first and even country the car was delivered to first or the VIN that tells me all that.

engine power?
battery power weak? lights dim, radio dies. etc, etc.etc.
transmission slips and car moving power (accel) less?
what. try to be more clear?
say what works and not. cars are very complex.l

Is the check engine light on ?
Usually when the engine is shaking you have a miss where one or more cylinders are not firing or not producing power.

Symptoms don't sound too serious and you arent in any danger from driving it, you are however in risk of damaging your engine. It sounds like a cracked transmission or your alternator isnt producing enough electricity for your car. Either way you should go to a mechanic unless you can do it yourself.

I would first try draining the carb bowl, if you look at where the choke lever is and look below that you should see a clear red tube with a flat head screw below that, if you just turn the screw out the carb will drain. If you still have an issue you are going to have to clean the carb.

The things to check first is the battery and charging system.
Quick check would be to use a Volt meter to measure across the battery, at the terminals. Should be at least 12vdc with engine off.
If less, may be a dead cell. Recharge or replace the battery.
Start engine, should now measure about 14 vdc. This would mean the charging system is working. Less than 14 vdc would mean low battery or problem with the rectifier parts.

check the battery then if low in electricity, then charge it if missing in Idle throttle then check and repair the fuel pump or carburetor assembly.when check engine lights on it means there is problem that is happening to the engine.like low fuel is going through the engine.and the power produced is less.

sound like to me you have a clogged catalytic converter

Sounds like the alternator needs replacing or a loose wire/cable.Check the starter and alternator wires/cables are securely fastened.Also use a multi meter to check wether your alternator is providing enough power to run the electrical system,should be producing no less than 12.8 with all lights etc switched on.

I would check all the connections, try switching different connections making sure it's not the jacks, if that doesn't work, try sending to company if under warranty.

hope this helps