Can your child be born with a cancer already growing inside him?

Yes. A few tumors, e.g. Teratomas, may be present at birth. Some childhood tumors that may present in a neonate could be presumed to have started before birth, e.g., neuroblastoma. Some vascular tumors. Though not cancerous, develop in utero and are present at birth.
While the incidence. Is tiny, lymphocytic leukemia, neuroblastoma, medulloblastoma, retinoblastoma and wilms can be present before birth. Some in-born errors in development are serious enough to cause incomplete development and in utero demise leading to spontaneous abortion (miscarriage to those not liking the medical term).
Yes. This isn't very common, but some cancers begin growing while in utero. Benign taratomas are the most common sited, but malignant tumors have been recorded.