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@jk2 wrote:

The child does not have a Social or Itin. 

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First, you didn't tell us whether your client was married and, if so, whether the spouse was a US person or NRA.  This is important because the §2(b)(2)(B) exception only applies if the spouse of a married taxpayer was an NRA - understand, however, your client being single is probably a foregone conclusion since you wouldn't have asked this question if your client could file MFJ.

For the sake of technical discussion, even if your client were to be in the Philippines and did live with the child for more than 1/2 of the year, your client would still not be eligible to file as HoH by virtue of failing the condition stipulated under §152(b)(3) - I am presuming the child was not a US citizen, national, or resident since the child does not have an SSN or ITIN.