This study examines the impact of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>C</a:mi><a:mi>P</a:mi></a:math>-violation on the signal strength <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>μ</c:mi><c:mrow><c:mi>Z</c:mi><c:mi>γ</c:mi></c:mrow></c:msup></c:math>, which was reported as <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mn>2.2</e:mn><e:mo>±</e:mo><e:mn>0.7</e:mn></e:math> by the LHC. We obtain constraints on the real and absorptive parts of the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>C</g:mi><g:mi>P</g:mi></g:math>-violating form factor <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msubsup><i:mi>h</i:mi><i:mn>3</i:mn><i:mrow><i:mi>Z</i:mi><i:mi>γ</i:mi></i:mrow></i:msubsup></i:math> and find that they are less than 1.15 GeV. Additionally, we revisit the leading order Standard Model (SM) contributions to the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>H</k:mi><k:mo stretchy="false">→</k:mo><k:mi>Z</k:mi><k:mi>γ</k:mi></k:math> decay and calculate contributions to <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msubsup><n:mi>h</n:mi><n:mn>3</n:mn><n:mrow><n:mi>Z</n:mi><n:mi>γ</n:mi></n:mrow></n:msubsup></n:math> from flavor-changing neutral current (FCNC) complex couplings mediated by the <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>Z</p:mi></p:math> and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>H</r:mi></r:math> bosons. By using the current bounds on such couplings, we find that the FCNC contribution to <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:msubsup><t:mi>h</t:mi><t:mn>3</t:mn><t:mrow><t:mi>Z</t:mi><t:mi>γ</t:mi></t:mrow></t:msubsup></t:math> with top and charm quarks in the loop is of order <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msup><v:mn>10</v:mn><v:mrow><v:mo>−</v:mo><v:mn>5</v:mn></v:mrow></v:msup><v:mtext> </v:mtext><v:mtext> </v:mtext><v:mi>GeV</v:mi></v:math>. While in a model with new quarks that preserves the SM predictions on Higgs decays, the <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi>C</x:mi><x:mi>P</x:mi></x:math>-violating form factor <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:msubsup><z:mi>h</z:mi><z:mn>3</z:mn><z:mrow><z:mi>Z</z:mi><z:mi>γ</z:mi></z:mrow></z:msubsup></z:math> can be of order <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:msup><bb:mn>10</bb:mn><bb:mrow><bb:mo>−</bb:mo><bb:mn>1</bb:mn></bb:mrow></bb:msup><bb:mtext> </bb:mtext><bb:mtext> </bb:mtext><bb:mi>GeV</bb:mi></bb:math> and could explain the excess on the signal strength <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:msup><db:mi>μ</db:mi><db:mrow><db:mi>Z</db:mi><db:mi>γ</db:mi></db:mrow></db:msup></db:math>. Published by the American Physical Society 2025