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The present paper investigates the following inhomogeneous generalized Hartree equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mover accent="true"><mi>u</mi><mo>˙</mo></mover><mo>+</mo><mo>Δ</mo><mi>u</mi><mo>=</mo><mo>±</mo><mrow><mrow><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><msup><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mi>b</mi></msup></mrow><mrow><mo stretchy="false">(</mo></mrow><msub><mi>I</mi><mi>α</mi></msub><msup><mrow><mo>∗</mo><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mi>p</mi></msup><msup><mrow><mo>|</mo><mo>·</mo><mo>|</mo></mrow><mi>b</mi></msup><mrow><mo stretchy="false">)</mo></mrow></mrow><mi>u</mi></mrow></semantics></math></inline-formula>, where the wave function is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>:</mo><mo>=</mo><mi>u</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>:</mo><mi mathvariant="double-struck">R</mi><mo>×</mo><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup><mo>→</mo><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>. In addition, the exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> gives an unbounded inhomogeneous term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mi>b</mi></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mi>α</mi></msub><mo>≈</mo><msup><mrow><mo>|</mo><mo>·</mo><mo>|</mo></mrow><mrow><mo>−</mo><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></msup></mrow></semantics></math></inline-formula> denotes the Riesz-potential for certain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mi>N</mi></mrow></semantics></math></inline-formula>. In this work, our aim is to establish the local existence of solutions in some radial Sobolev spaces, as well as the global existence for small data and the decay of energy sub-critical defocusing global solutions. Our results complement the recent work (Sharp threshold of global well-posedness versus finite time blow-up for a class of inhomogeneous Choquard equations, J. Math. Phys. 60 (2019), 081514). The main challenge in this work is to overcome the singularity of the unbounded inhomogeneous term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mi>b</mi></msup></semantics></math></inline-formula> for certain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>.