The assessment of children at risk of autism typically involves a clinician observing, taking notes, and rating children's behaviors. A machine learning model that can label adult and child audio may largely save labor in coding children's behaviors, helping clinicians capture critical events and better communicate with parents. In this study, we leverage Wav2Vec 2.0 (W2V2), pre-trained on 4300-hour of home audio of children under 5 years old, to build a unified system for tasks of clinician-child speaker diarization and vocalization classification (VC). To enhance children's VC, we build a W2V2 phoneme recognition system for children under 4 years old, and we incorporate its phonetically-tuned embeddings as auxiliary features or recognize pseudo phonetic transcripts as an auxiliary task. We test our method on two corpora (Rapid-ABC and BabbleCor) and obtain consistent improvements. Additionally, we outperform the state-of-the-art performance on the reproducible subset of BabbleCor. Code available at https://huggingface.co/lijialudew