my $urlWithSpaces = 'C:/Code/Perl/Phi Kap/temp.html';
start qq["$url"];
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Where are you getting the bareword 'start' to do something? You also are using two difference variable names. I just get

syntax error at - line 2, near "start qq["$url"]"
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Perhaps you meant:

my $url = 'C:/Code/Perl/Phi Kap/temp.html';
system( qq[start "Title" "$url"] );
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(Note the inclusion of "Title" to tell start that $url isn't a window title desptie it being inside of double quotes.)

Or perhaps you left off some incantation that makes the code you wrote work?

(Updated)