I hope I did understand your problem. Here is my sample code. Kindly check it if the flow and output fits to the solution of your problem. If it didn't, I hope it can give you some clues. Just like Javafan said check each word of "my_name" if they all exist in one of the string of database "db_name".

#!/usr/bin/perl -w

@my_name=("Acidovorax","JS42");    #name to be searched...
                                   #...split by term/words
@db_name=("Acidovorax sp. JS42",    #data base of names in array
          "JS42 Acidovorax sp.",    
          "JS42 sp. Acidovorax",
          "JS53 sp. Acidovorax",
          "JS42 sp. Axidovorax",
          "JS42Acidovorax sp. "
          );

my $ctr;
#----compare strings in $db_name 1 at a time
foreach my $db_each (@db_name){            
   
   #----search each term/word of @my_name in $db_each
   foreach($ctr=0; $ctr<=$#my_name; $ctr++)   
   {
          #----if a term/word not found break the loop
       last if($db_each !~ /\b$my_name[$ctr]\b/i);  
   }                                                
   #----this will be true if inner foreach didn't break
   if($ctr==$#my_name+1)                            
   {
       print "$db_each\n";       #print the matched name
   }
}
<>;
[download]

I have mention the "term/word" in the comment. It might be that two words in your "my_name" is considered as one term or two or more words separated by space.

The code above does not support if a word should exist 2 or more times in any order. example: my_name = "high class bacteria f7-52 high fever". Just mentioning this condition for more flexibility to your program. It's still up to you.

If I change the assignment to @my_name to

@my_name=("Acidovorax", "sp.", "JS42");
[download]

the program fails to find a single match. I doubt this is want the OP wants. The problem is that you're using different "word" definitions for "my_name" and "db_name". Either split both on whitespace, or both on \b, or it becomes trivial to find examples where the search is going to return the wrong answer. For instance:

use 5.10.0;
use strict;
no warnings;

my @names    = ("Acidovorax JS42", "Acidovorax sp. JS42");
my @db_names = ("Acidovorax sp. JS42",    #data base of names in array
                "JS42 Acidovorax sp.",
                "JS42 sp. Acidovorax",
                "Acidovorax JS42",
                "JS53 sp. Acidovorax",
                "JS42 sp. Axidovorax",
                "JS42Acidovorax sp. "
                );

foreach (@names) {
    say;
    my %name; @name{+split} = ();
    foreach (@db_names) {
        my %copy = %name;
        delete $copy{$_} for split;
        say "\t$_ is ", (keys %copy ? "not a " : "a "), "match";
    }
}
__END__
Acidovorax JS42
    Acidovorax sp. JS42 is a match
    JS42 Acidovorax sp. is a match
    JS42 sp. Acidovorax is a match
    Acidovorax JS42 is a match
    JS53 sp. Acidovorax is not a match
    JS42 sp. Axidovorax is not a match
    JS42Acidovorax sp.  is not a match
Acidovorax sp. JS42
    Acidovorax sp. JS42 is a match
    JS42 Acidovorax sp. is a match
    JS42 sp. Acidovorax is a match
    Acidovorax JS42 is not a match
    JS53 sp. Acidovorax is not a match
    JS42 sp. Axidovorax is not a match
    JS42Acidovorax sp.  is not a match
[download]

use strict;
use warnings;

my $my_name="Acidovorax JS42";
my $db_name="Acidovorax sp. JS42";

my @names = split " ", $my_name;
my $numel = @names;  #number of words to match

my $count;
foreach (@names) {
  if ($db_name=~/^|\s\Q$_\E$|\s/) {
     $count++;  #count matches
  }
}

print "MATCH" if $count==$numel;  #report match if all words match
[download]

You would have to escape $_ as it may contain regexp special characters. It also assumes "words" are delimited by whitespace and by \b at the same time - your program fails to find a match if $my_name=$db_name="Acidovorax sp. JS42" for instance.