I wasn't familiar with the ASC-H category, so I did some research and found that it means 'Atypical squamous cells- cannot exclude HSIL'. Evidentially, it means that there are some changes in the cervical cells that are suggestive of HSIL, but these changes aren't suffient enough to actually be classified as HSIL.
That may be a good thing, but I've found that sometimes Paps can be totally off mark when it comes to accurately grading pre-cancerous lesions. For instance, there have been ladies whose Paps were high grade, but whose biopsies came back low grade or just ASC-US. On the flip side, my Paps were showing ASC-US, when in fact I had HSIL; and two years later, when my Pap showed LSIL, I actually had carcinoma in situ (CIS).
Paps really are only a sampling of a few cells, and until a biopsy is done, there is no real way of knowing what is going on.
Since cervical cancer is so slow to develop, a delay of a few weeks or month or so shouldn't make much of a difference. For instance my LSIL Pap was in November 2006, I didn't have the colposcopy and biopsy which showed the CIS until February 2007, and the laser surgery for the CIS wasn't until May 2007.
Good luck with everything and please keep us updated on how things are going. We're here whenever you need us.