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No. No hydrogen attached so it can't form H-Bonds.
But, it does make the hydroxyl bond more polar. This increases the strength of hydrogen bonds in the substance. For your question, you just have to see which substituants will act the most like a carbonyl group. Since C=O is e- with drawing, and a Cl substituant is also withdrawing, they will have the most similar BP.
Carboxylic acids can definitely engage in hydrogen bonding, especially the C=O oxygen. Carboxylic acids may form dimer-like structures by hydrogen bonding between multiple molecules of the same acid. This gives carboxylic acids a higher boiling point relative to alcohols.
edit: Oh haha I misread CH3COH as an alcohol. After reading sleepy's post below I now see that it is an aldehyde. This question made no sense if CH3CH2OH and CH2ClCH2OH were being used as the comparisons.
They definitely can. CH3COOH molecules' C=O and OH have somewhat complementary polarities allowing them to form dimers with each other via h-bonds, which makes the BP pretty high, even higher than H2O. It looks like that question wants you to pick an answer where the molecule has an EWG that has the same effect as a C=O on h-bonding.
the carbonyl oxygen can function as a hydrogen-bond acceptor. it cannot function as a hydrogen bond donor.
acetic acid (CH3COOH) can form hydrogen bonds with itself because the carbonyl oxygen (and the other oxygen) can function as a hydrogen bond acceptor, and the O-H bond functions as a hydrogen bond donor so hydrogen bonding plays an important role in the determination of the boiling point of acetic acid
acetyaldehyde (CH3COH) is an aldehyde, not an alcohol. Just because it says OH doesn't mean it's an alcohol (actually, i think you noticed that based on your question, but I just wanted to emphasize that). acetaldehyde cannot form hydrogen bonds with itself (well, with other molecules of itself, i.e. dimerization) because it can only function as a hydrogen bond acceptor. if you were to put it in water, it could hydrogen bond with water (which is why it's water soluble), but it cannot hydrogen bond with another acetaldehyde molecule.
2-chloroethanol can hydrogen-bond with other molecules of 2-chloroethanol because the oxygen can function as a hydrogen-bond acceptor and the O-H bond can function as a hydrogen-bond donor.
that's why the boiling points of 2-chloroethanol and acetic acid are reasonably close.
so remember, for a pure substance to exhibit hydrogen-bonding, it must be able to function as both a hydrogen-bond donor and as a hydrogen bond acceptor.