Use a hash to check for hits from the smaller data set (A in the sample) and generate a composite list:

use strict;
use warnings;

my @DataA = split /\n/, <<DATAA;
A   "Monday"
B   "Tuesday"
C   "Wednesday"
D   "Thursday"
DATAA

my @DataB = split /\n/, <<DATAB;
M    252   212  "Bill"
A    325   908  "Jim"
C    426   907  "Mike"
A    423   383  "Sally"
A    993   421  "Jim"
C    737   432  "Mary"
DATAB

my %AHits;
my @merged;

for (@DataA) {
    my ($key, $day) = split /\s+/;
    
    $AHits{$key} = $day;
}

for (@DataB) {
    my ($key, @params) = split /\s+/;
    
    next unless exists $AHits{$key};
    
    push @merged, [$key, $AHits{$key}, @params];
}

print join ("\t", @$_), "\n" for sort {$a->[0] cmp $b->[0]} @merged;
[download]

Prints:

A    "Monday"    325    908    "Jim"
A    "Monday"    423    383    "Sally"
A    "Monday"    993    421    "Jim"
C    "Wednesday"    426    907    "Mike"
C    "Wednesday"    737    432    "Mary"
[download]