'$x$' is the product which is obtained by the hydrolysis of prop-1-yne in the presence of mercuric sulphate under dilute acidic medium at $333$ K. '$y$' is the product which is obtained by the reaction of ethane nitrile with methyl magnesium bromide in dry ether followed by hydrolysis. IUPAC name of product obtained from '$x$' and '$y$' in the presence of barium hydroxide followed by heating is :

Hydrolysis of prop-1-yne in the presence of $HgSO_4$ and dilute $H_2SO_4$ yields acetone.

$CH_3-C \equiv CH + H_2O \xrightarrow{Hg^{2+}, H^+} CH_3-CO-CH_3$

Thus, $x$ is acetone.

Reaction of ethane nitrile ($CH_3CN$) with methyl magnesium bromide ($CH_3MgBr$) followed by hydrolysis also yields acetone.

$CH_3-C \equiv N + CH_3MgBr \rightarrow CH_3-C(CH_3)=NMgBr \xrightarrow{H_3O^+} CH_3-CO-CH_3$

Thus, $y$ is acetone.

Reaction of $x$ and $y$ (both acetone) in the presence of $Ba(OH)_2$ followed by heating results in an aldol condensation.

$2 CH_3-CO-CH_3 \xrightarrow{Ba(OH)_2} CH_3-C(OH)(CH_3)-CH_2-CO-CH_3$

Heating causes dehydration to form an unsaturated ketone.

$CH_3-C(OH)(CH_3)-CH_2-CO-CH_3 \xrightarrow{\Delta} CH_3-C(CH_3)=CH-CO-CH_3$

Numbering the principal carbon chain to give the lowest locant to the ketone group, the IUPAC name of the product is 4-methylpent-3-en-2-one.

Answer: 4-Methylpent-3-en-2-one