No, it isn't entirely correct grammatically. Let's simplify for a minute, by putting the subordinate clause last:
It seems that the earnings of the low-paid classes will diminish, what with/on account of an increased rate of inflation and the public aid having been eliminated.
First, you have to say the public aid having been eliminated rather than having been eliminated the public aid. Consider this:
The public aid has been eliminated.
We have eliminated the public aid.
You'll note that the word order for passive voice is the reverse of that for active voice.
Next, on account of/what with refers to both the public aid and the inflation, so you need to reflect that by putting it before both ideas.
Now, you can put the subordinate clause first for emphasis, as you have:
What with/on account of the public aid having been eliminated and an increased rate of inflation, it seems that the earnings of the low-paid class will diminish.
Now, which to use, on account of or what with? I lean towards what with, because it has more of a flavor of speculation. When you say "it seems that" that also suggests speculation, so I find it a bit more in keeping with the overall speculative idea of the sentence. Either is entirely correct, though.