Apple - Is there a way to make Siri listen properly?

As far as I'm aware there is no way to get around this if you want to continue to use Siri from the lock screen or by holding the home button.

Your only solution at the minute is either to make sure you say everything without pausing or enable and use the dictation feature on the keyboard - It listens indefinitely until you tap on 'done' so you will be able to pause to think without getting cut off.

Just continue to hold down the home button (or Siri microphone button) after Siri has been activated and you have started talking.

If you do this, Siri will wait until you let go of the button to process what you have said, so you can make pauses to your heart's content without triggering the "I've stopped" action.

No - Siri currently has no way to program the pauses to detect when you are done speaking. You could have siri open notes and then use the dictation feature to dictate your thoughts with pauses using the keyboard command and then edit them and then finally push that composed thought to messages, but it defeats the ease of using siri to automate the sending of your message.

Without knowing your context (Are you driving and totally hands free? Are you without function of arms/fingers either temporarily or permanently?) it's hard to see if there's a solution to your end goal, but your short term solution is to change the way you interact with Siri to leave less pause. Perhaps just skipping the formality of "Hi John" and just blurt out what you need John to know/do.

Once you've trained yourself to get the main message sent you can work on tacking on "comma thanks so much exclamation point" (or period) or whatever you wish to humanize the main point that's about to be sent.