Its been months now that our water pressure is not enough to fully flush the commode, every time it must be filled manually or wait until it fills for the second time for a complete flush out.
Flush toilets with the usual non-pressurized water tanks don't need significant water main pressure to function. The depth of the water in the tank is what provides the pressure. Low water main pressure will increase the time to fill the toilet tank, but once it's full the flush process will be the same.
In your case, it's likely there's something wrong with the toilet flushing mechanism, or the drain pipe.
Improper flushing often results from minerals clogging up the holes under the rim of the toilet. The usual cure is to clear out the buildup with a muriatic acid treatment.
I agree with Daniel Griscom. Get a Plumber or a new toilet or both to save yourself from the below ordeal, which isn't bad it's just experience. The plus is that you'll be able to unclog most any toilet thereafter.
As long as the Cistern Tank's dumping its water quickly, then you've got a slow draining problem. A slow draining tank is extremely rare & the only solution would be to replace everything in & attaching the tank. Otherwise, floss the toilet's entire very short run of drain, which you may see embossed on the side of the toilet's bottom section under the tank.
You want to get a Drain Auger or turned on Garden Hose into every portion of the toilet's drain while repeatedly flushing...poke, twist & bend with the flushing to hopefully catch the edge of whatever debris. If this has only a slight improvement, then you'll need to remove the toilet & do the same from the other end into the toilet bowl to ensure the toilet's drain is clear.
Finally, one main cause of clogs & slow draining is the toilet's connecting pipe & thick wax seal. These can get built-up with debris & that will all need to be removed from the pipe's connecting end with a putty knife. Your hand should fit into the pipe. If the pipe is cast iron, the you'll need to chisel any restricting rust off to full pipe diameter while using a vacuum to retrieve the rust you remove.