Is it possible to change mercury's specific weight to make it heavier? By mixing it with anything? But not by changing its temperature.
Specific weight is a property of the element. It is product of the atomic mass and the volume. You can't easily alter either.
You could mix an element with another element of a different density and the density of the mixture will change. But you won't be changing the specific mass of the elements, just creating a mixture with a different density.
You can't readily change the mass of an element. But an element might be made up of multiple isotopes (mercury has 4 significant ones with masses ranging from 199 to 202 atomic mass units). If you enrich the mercury for one of those isotopes, you can change the mass, very slightly (for other elements this can be important; fissile isotopes of uranium are concentrated by doing it when making nuclear fuels).
In practice there is nothing practical you can do to alter the density (specific mass) of an element though you can create mixtures with different densities (but then you still haven't altered the properties of the element itself).