#!/usr/bin/perl
my $line;
while (my $line = <DATA>){
if($line =~ m/<span><p>(.*)/){
$line= $1;
$line =~ s/</<\/p><p>/g;
print $line;
}
}

__DATA__
<span><p>This affected me deeply because < I was being treated for dep
+ression then. < felt she was replacing me as mother and grandmother.
[download]

This affected me deeply because </p><p> I was being treated for depres
+sion then. </p><p> felt she was replacing me as mother and grandmothe
+r.
[download]

<span><p>This affected me deeply because </p><p> I was being treated f
+or depression then. </p><p> felt she was replacing me as mother and g
+randmother.
[download]

Consider this instead if your idea is to capture text between any kind of html start and end tag pair.

if($line =~ m/<.+>(.+)<\/.+>/){ 
  print $1; 
}
[download]

UPDATE: I see now that the question i replied to has been replaced by you ... probably updated. So my answer above doesn't hold.

If I use substitution of $1, then <span><p> is removed from the input data which printing the output. In the script

#!/usr/bin/perl
my $line;
while (my $line = <DATA>){
if($line =~ m/<span><p>(.*)/){
$line = $1;
$line =~ s/</<\/p><p>/g;
print $line;
}
#}
}
__DATA__
<span><p>This affected me deeply because < I was being treated for dep
+ression then. < felt she was replacing me as mother and grandmother.
[download]

From the above code the output is below.

This affected me deeply because </p><p> I was being treated for depres
+sion then. </p><p> felt she was replacing me as mother and grandmothe
+r.
[download]

But I have to print the line, such that it does substitution

<span><p>This affected me deeply because </p><p> I was being treated f
+or depression then. </p><p> felt she was replacing me as mother and g
+randmother.
[download]