Can a parent get their child back after a temporary care and custody agreement expires?

Generally, yes. A temporary care and custody agreement is meant to be time-limited by law, and once it lapses without being renewed or replaced by a court order, the society has no ongoing legal authority under that agreement to keep your child from returning to your care. If the society still believes the child needs protection at that point, it's expected to apply to the child protection court before the agreement ends, rather than simply holding onto the child informally or refusing to hand them back.

Before the end date arrives, confirm in writing what happens next: is the child being returned, is the society proposing a renewal, or is it planning to start a court application? If the society tries to keep the child past the agreement's end without your consent and without a court order, that's a problem worth raising with a lawyer immediately - the society's authority under the agreement doesn't continue on its own past the term you agreed to. If you already have a separate family court parenting case, tell the other parent's counsel and your own lawyer about the return, since it can affect scheduling and any interim arrangements.