Do Mormons Believe in a Heavenly Mother?

Yes.  Mormon dogma does allow for the existence of a Heavenly Mother or female counterpart to a fully embodied and physical Heavenly Father.  However, the existence of a Heavenly Mother is not mentioned in the LDS canon of scripture and is purely a dogmatic assertion based on both canon and official doctrine.  The most explicit references to Her existence can be found in the LDS hymn “Oh My Father” and in the recent proclamation which makes mention of “heavenly parents.”

Some LDS feminists have advocated for a more prominent role and more open discussion of the Heavenly Mother and in some cases have advocated praying directly to Her, rather than to the Father.  The most prominent LDS feminists have argued that the Holy Ghost is in fact, the Heavenly Mother.  Not surprisingly such claims caused significant tension between these feminists and the Church institution and Church leaders came out very forcefully stressing that prayer should only be addressed to the Father and that speculations about the Heavenly Mother are inappropriate. 

The bottom line is that there is absolutely no canon support for the existence of the Heavenly Mother.  Late speeches of Joseph Smith (such as the King Follett Discourse) imply Her existence but do not do so explicitly.  Thus, any discussion of the Heavenly Mother are inherently speculative.