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Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>M</mi><mn>1</mn></msub><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>M</mi><mn>2</mn></msub><mo>,</mo><mi>h</mi><mo>)</mo></mrow></semantics></math></inline-formula> be two Hermitian manifolds. The twisted product Hermitian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>M</mi><mn>1</mn></msub><mo>×</mo><mmultiscripts><mi>M</mi><mn>2</mn><mi>f</mi></mmultiscripts><mo>,</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the product manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mn>1</mn></msub><mo>×</mo><msub><mi>M</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> endowed with the Hermitian metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mi>g</mi><mo>+</mo><msup><mi>f</mi><mn>2</mn></msup><mi>h</mi></mrow></semantics></math></inline-formula>, where <i>f</i> is a positive smooth function on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mn>1</mn></msub><mo>×</mo><msub><mi>M</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>. In this paper, the Chern curvature, Chern Ricci curvature, Chern Ricci scalar curvature and holomorphic sectional curvature of the twisted product Hermitian manifold are derived. The necessary and sufficient conditions for the compact twisted product Hermitian manifold to have constant holomorphic sectional curvature are obtained. Under the condition that the logarithm of the twisted function is pluriharmonic, it is proved that the twisted product Hermitian manifold is Chern flat or Chern Ricci-flat, if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><msub><mi>M</mi><mn>1</mn></msub><mo>,</mo><mi>g</mi></mfenced></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><msub><mi>M</mi><mn>2</mn></msub><mo>,</mo><mi>h</mi></mfenced></semantics></math></inline-formula> are Chern flat or Chern Ricci-flat, respectively.