Hydrogen Bonding (with oxygen) - restricted to oxygen singly bonded to H or not?
Hydrogen-bonds require a $\delta^+$ hydrogen atom attached to an electronegative element, i.e. O, F, N, or Cl, and also an "active" lone pair on another electronegative atom.  An electrostatic attraction between these two is an extremely strong kind of intermolecular dipole-dipole interaction.

Compounds which have O-H or N-H groups are able to both form hydrogen-bonds through the $\ce \delta^+$H atom and accept hydrogen-bonds through the lone pair on the oxygen or nitrogen.  These compounds tend to have higher then expected melting and boiling points due to the hydrogen-bonding.  Compounds are even able to form internal hydrogen-bonds with other O or N atoms in the structure.

However hydrogen bonding can occur between substances that cannot form the bonds with their own molecules.  Because compound Y is an ether and can only accept hydrogen-bonds there are fewer bonds made overall between the mixture of substances, so the effect on the physical properties isn't so great.  Nonetheless water can form hydrogen-bonds through its proton and functional groups such as ethers, esters, amines, amides, alcohols and carboxylic acids can all accept those bonds through the relevant lone pairs.

Another example would be a tertiary amine such as $\ce{N(CH3)3}$, which is miscible with water because of the hydrogen bonds it accepts from the water molecules even though it has no N-H bonds itself.  Even $\ce{N(C2H5)3}$ is miscible with water at room temperature, and other tertiary amines continue to be soluble in water for the same reason: they are hydrogen-bond acceptors.

So the mark scheme is correct, and the teacher has shown a poor understanding of intermolecular interactions.