Operation with second-order logic formulas I have one or more second-order logic formulas that correct for some finity model. What operation can I perform on them, so that resault is also correct for some finity model?

Just conjunction is impossible, example:
$x$ has finity model, $\lnot x$ has finity model, but $x \wedge \lnot x$ has not finity model. Are there any restriction?
Out of the standard logical operations ($\lor$, $\land$, $\neg$, $\implies$), the only ones that meet your conditions are $\lor$ and $\implies$.