In any case, parents should make clear what, if anything/if any, the child is expected to pay (for something) with the pocket money.
In the sentence above, which is more acceptable? What is the difference between the two phrases?
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"If anything" is more acceptable.
In the phrase "if any", 'any' is a determiner. Since the "what" is not established, "any" does not have an implied noun to go after it.
The sentence could be rewritten to more comfortably use "if any" in the following way:
In any case, parents should make clear what things, if [there are] any [things], the child is expected to pay for with the pocket money.
An example of using "What, if any" that doesn't sound bad:
What, if any, are the effects of direct sunlight on human eyes?
When used alone (without a noun after it), "any" is a pronoun just like "anything". The main difference is that "any" will refer to a noun that is mentioned in the context, while "anything" is, as you surely know, formed by "any" + "thing" and, as such, already contains a noun, though the noun is indeterminate (that is why "anything" is known as an indefinite pronoun).
In the sentence at issue, there is no noun but the nominal relative pronoun "what", which may be broken down into "what" + "thing". Though implicit, the noun "thing" is not mentioned, and therefore we need "if anything" to complete that space.
In any case, parents should make clear what, if anything, the child is expected to pay for with the pocket money. (In this case, "anything" can refer to any unidentified object or service which the child is expected to pay for, and is as indeterminate as "what". This is why "anything" can form a pair with "what" -- they are both indefinite as to the identitiy of the thing in question.)
In any case, parents should make clear what leisure activities, if any, the child is expected to pay for with the pocket money. (In this case, "any" refers to the noun "leisure activities", the meaning being: "if he is actually expected to pay for any leisure activities with the pocket money." Since there is a noun in the context, "anything" cannot be used because there would be an overlap. However, we can find this:)
In any case, parents should make clear what leisure activities, if anything at all, the child is expected to pay for with the pocket money. (In this case, "leisure activities" and "anything at all" do not clash, because the latter includes any other things -- aside from leisure activities -- which the child should or should not buy with his pocket money.)