If it flushes well by bucket method,snaking is probably not what is needed. Do you have long running experience with the toilet where it flushed well? There was a stretch of time when low consumption toilets irst came into use that they were poor preformers.

It is newly installed..a handicapped height and it has Styrofoam inside the tank..I am wondering if I cut out the styrafoam would it give me more water to in fact flush properly....thank you..I can't afford a plumber right now.

Maybe you had a blockage in sewer line that caused you to replace the toilet in the first place??????????
If the foam is indeed part of the desighn,I wouldn't alter it just yet. Does the foam line the walls to prevent tank sweating? I can't think of any other reason for foam so make certain it's not packing to prevent shipping damage to tank components. Could a piece of foam broke off and become traped between tank and trap? Not knowing what kind of system you have makes it difficult giving advise but let's try finding the problem. Before flushing, can you raise water level in tank pouring with your bucket? If the fill valve is not adjusted to supply enough water,that might tell us. The inverse can be true where tank supplys too much water and deminishes force o flush. Idea setting is when flapper falls closed shortly after water begins leaving bowel forcfuly. A slight possibility is that bowel was poorly alighned with floor flange or access wax squeezed out and partly blocked floor flange. During flush on many toilets,there is as much water flowing from bowel straight through flange as is leaving the bowel flowing through flange. With partialy blocked flange,there might be room to empty bowel but insuffecent to carry the tank flow as well.

I do appreciate your effort in trying to help me..I have never asked for advise before online..nice to know there are good people out there...anyway...I added water to tank all the way up to overflow pipe thingy and it swirled but would not flush..then poured bucket into commode and flushed so easy...it was replaced for the added handicapped height for me. Ovarian cancer has taken my body strength and it is amazing how much the extra inches help the pain ..the other toilet worked fine.....

Can you take a picture of the styrofoam? I too am wondering if that is left over packing, and something got caught between the tank and toilet (therefore, not plugging the holes, per se, but reducing the water flow between tank and bowl).

Probably not what you want to hear but from here it looks like hands on for someone knowledgable about toilets. You may have a defective toilet or unusual issue. This is a new toilet????? I understand about expense of plumbers and have a couple of suggestions that might spare some expense.
If you hired a plumber to install it;
They should have tested to make certain the toilet worked properly after install. Did it work right,and for how long? It costs nothing to ask for a no charge or reduced price service call. If that can't be arranged,ask at your church and sr citizen center if someone volinteers through them. Personaly I always considered gratis work for those less fortunate to be time well spent.
Failing other attempts to get help within your budget,try and track down an appartment maintainance person or childcare maintainance tech to see what they charge to have a look. I might hesitate useing those guys for some things but both have extensive experience with toilets.
Best of luck and sorry we didn't get to the problem but anything farther I might offer would likly be more than you want to tackle.

Thank you for your effort. and your suggestions .....the tank is new and the interior is lined with Styrofoam and even has a shelf on the right side...a water saver toilet.....stupid...
be well