The reason for Sarah and Abraham having a child with Hagar, Sarah's maidservant, is primarily attributed to Sarah's barrenness and her desire to fulfill God's promise of descendants to Abraham. In the biblical narrative found in Genesis 16, Sarah, who was unable to conceive, suggested to Abraham that he should have a child with her Egyptian maidservant, Hagar, so that they could raise the child as their own.
Sarah likely saw this as a way to ensure the continuation of Abraham's lineage and the fulfillment of God's promise to make him the father of many nations, as he had been told by God earlier in the book of Genesis. However, the decision to take matters into their own hands and use Hagar as a surrogate ultimately led to complications and conflict within their household.
After Hagar conceives and bears a son named Ishmael, tensions arise between her and Sarah, leading to mistreatment and eventually Hagar's departure from Abraham's household. Despite Ishmael being born as a result of Sarah and Abraham's attempt to fulfill God's promise through human effort, God later reaffirms his covenant with Abraham, promising that Sarah herself will bear a son, Isaac, who will be the child of promise. This underscores the theme of divine sovereignty and the importance of trusting in God's timing and faithfulness rather than relying on human schemes to accomplish his promises.