How do I get rid of the unwanted line shift between the second and third line?
\documentclass[12pt]{book}
\usepackage{verse}
\PassOptionsToPackage{bookmarks, colorlinks=false, hidelinks}{hyperref}
% Use PoetryTeX; http://www.ctan.org/pkg/poetrytex
\usepackage[numberpoems, clearpageafterpoem, useincipits]{poetrytex}
% Use the PA5 paper size
\usepackage[paperwidth=140mm,paperheight=210mm]{geometry}
\begin{document}
\begin{poem}{~}
\poemtitle{Poem name}
\settowidth{\versewidth}{And objects at rest tended to remain at rest tended to remain at rest}
\begin{verse}[\versewidth]
This is the first line\\
\mbox{The second line is much longer than the first} \\
\mbox{The third has unwanted space from the second}
\end{verse}
\end{poem}
\end{document}

\mbox{The third has unwanted space from the second}(I removed it from your MWE). – Skillmon Jan 04 '20 at 20:12\\on the second line, after the\mbox. Since that line is longer than the default width, the space would cause that line to go to a second output line. – barbara beeton Jan 04 '20 at 20:31