<p>Modified Korteweg-de Vries equations are used to describe certain phenomena in nonlinear optics, fluid dynamics, plasma physics, ocean physics and gas dynamics. In this paper, we investigate a damped variable-coefficient fifth-order modified Korteweg-de Vries equation for the small-amplitude surface waves in a strait or large channel of slowly-varying depth, width and non-vanishing vorticity. Via the truncated Painlevé expansion, Painlevé-type auto-Bäcklund transformations are obtained. Based on the Painlevé-type auto-Bäcklund transformations, periodic solutions and one-soliton solutions are derived. The soliton-like solutions are constructed via the modified Kudryashov method. We graphically show the kink-type solitons of the soliton solutions. Graphic analysis shows that the shapes, characteristic lines and velocities of the solitons are related to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2024_1196_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha _{1}(t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2024_1196_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta (t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, while the soliton backgrounds and amplitudes just depend on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2024_1196_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\gamma (t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, in which <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2024_1196_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha _{1}(t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2024_1196_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta (t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2024_1196_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\gamma (t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are the dispersive, dissipative and line-damping coefficients of the equation we investigated, respectively, where <i>t</i> is the temporal variable.</p>