Here's a minimal example that shows one way to do it (probably the simplest way):

#!/usr/bin/perl

use strict;
use Tk;

my $mw = MainWindow->new();

my $txtvar;  # have a separate scalar for each Entry widget

$mw->Entry(-textvariable=>\$txtvar # this is the trick
           )->pack();

$mw->Button(-text=>"Print & Clear",
            -command=>sub{print $txtvar,$/; $txtvar=""}
            )->pack();

MainLoop;
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If the Entry widget is configured to have a "-textvariable" (a reference to a scalar), then the callback function on a button simply needs to reset that variable to an empty string. Naturally, if you have a lot of Entry widgets, it may be handy for their textvariables to be elements of an array or hash, in which case you give each Entry a reference to an element of the array or hash:

my %strings = qw/first one second two third three/;

for my $name ( sort keys %strings ) {
    $mw->Label(-text => $name)->pack();
    $mw->Entry(-textvariable => \$strings{$name})->pack();
}

$mw->Button(-text=>"Print&Clear",
            -command=>sub{ for (sort keys %strings) {
                              print "$_=$strings{$_} ";
                              $strings{$_} = "";
                           }
                           print $/;
                         }
            )->pack();
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