Since you have had no replies I will attempt an answer, but expect to be corrected.
I would say yes. HB and CTB is an income based benefit. It makes no differnce how they get that income. It may be through work, other benefits etc. If they can prove that they did not have any income for that week then I would say you have to award benefit. Of course they will need to provide a letter from the employer saying that they had no income, and all other proof that a normal claimant would need to provide.
Of course they would need to apply before the week of no income, otherwise the date of claim would be the Monday following the receipt of the form. Since they are only claiming for one week, if they provided proof either in the week, or after the week the start date of the claim would be after the week of no income.