This doesn't do anything useful:

'div.abstract-content > p' || 'not provided'
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The || operator is a logical short circuit. If the expression on the left is true, that expression's value is returned, and the right-hand side is never evaluated. The string "div.abstract-content > p" is just a plain old string, and evaluates to true. Therefore, the 'not provided' string is never seen. You're not providing an alternate of choice, you're providing only one string as an option.

I'm guessing here but perhaps the path div.abstract-content > p doesn't resolve to anything in the DOM you are parsing. It might be useful for you to provide a sample of the dom, trimmed to show the relevant hierarchy.

Take a look at the Mojo::DOM documentation; especially the part regarding the at method, whose first sentence is:

"Find first descendant element of this element matching the CSS selector and return it as a Mojo::DOM object, or undef if none could be found."

Your posted error message reflects what the documentation describes; i.e. $dom1->at('div.abstract-content > p') is returning undef. I can reproduce this:

$ perl -MMojo::DOM -E 'my $dom = Mojo::DOM::->new("<p></p>"); my $x = 
+$dom->at("p.missing")->text(); say defined $x ? "not missing" : "real
+ly missing"'
Can't call method "text" on an undefined value at -e line 1.
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In order to "catch the exception", you should get the return value from $dom1->at('div.abstract-content > p'). If it is defined, you can use it to invoke the text() method; otherwise, handle as appropriate for your application (output a warning, write to a log, etc.). Based on my code above, you'd want something like this:

$ perl -MMojo::DOM -E 'my $dom = Mojo::DOM::->new("<p></p>"); my $x = 
+$dom->at("p.missing"); say defined $x ? "not missing" : "really missi
+ng"'
really missing
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Update (additional example): Something of an afterthought and really just intended to show the validity of the previous example:

$ perl -MMojo::DOM -E 'my $dom = Mojo::DOM::->new(q{<p class="missing"
+></p>"}); my $x = $dom->at("p.missing"); say defined $x ? "not missin
+g" : "really missing"'
not missing
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This is only an educated guess here, as I don't have the facilities currently to test, nor the time to read docs to understand what's happening. Can you extract the object, check for defined, then act appropriately?

Instead of:

my $abstr = $dom1->at('div.abstract-content > p' || 'not provided')->t
+ext;
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Will something like this example work?:

my $obj   = $dom1->at('div.abstract-content > p' || 'not provided');
my $abstr = defined $obj ? $obj->text : 'Object undefined';
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If it does work, you could elaborate on it a little, to say log the entries that are broken.

To the OP: Without having the time to test myself at the moment, what stevieb suggested should work; you can also choose a different action to take with a more complex statement like if ( defined $obj ) { ... } else { ... }. Note that 'div.abstract-content > p' || 'not provided' is actually not doing anything useful, as it's just a logical or and the first value is always true, so the second value will always be ignored.