**Core Concept**
The child's symptoms suggest a deficiency in a coagulation factor that affects both primary and secondary hemostasis, leading to excessive bleeding from minor injuries and joint effusion due to recurrent hemarthrosis.

**Why the Correct Answer is Right**
The clinical presentation of excessive bleeding from minor injuries, bleeding gums, and joint effusion in a young child with a family history of a bleeding disorder points towards Hemophilia A. Hemophilia A is caused by a deficiency in factor VIII, a critical component of the intrinsic coagulation pathway. Factor VIII is responsible for amplifying the coagulation cascade, ensuring proper clot formation. The child's symptoms, including joint effusion, are classic manifestations of Hemophilia A due to the spontaneous rupture of small blood vessels and subsequent bleeding into joints.

**Why Each Wrong Option is Incorrect**
* **Option A:** Factor XIII deficiency would not typically present with joint effusion or bleeding gums in a young child. Factor XIII is involved in stabilizing the blood clot, but its deficiency is more associated with skin and wound healing issues.
* **Option B:** Factor XI deficiency is another bleeding disorder, but it is less likely to cause joint effusion and more commonly presents with bleeding after surgery or trauma. Additionally, Factor XI deficiency is more common in Ashkenazi Jews.
* **Option C:** Factor VII deficiency is a bleeding disorder, but it is less likely to cause joint effusion and more commonly presents with bleeding after surgery or trauma. Factor VII is involved in the extrinsic coagulation pathway.

**Clinical Pearl / High-Yield Fact**
A patient with Hemophilia A is at risk of developing chronic joint disease due to recurrent hemarthrosis, leading to joint deformities and reduced quality of life. Early diagnosis and treatment with factor VIII replacement therapy are crucial to prevent long-term complications.

**Correct Answer:** D. Factor VIII deficiency.