Finding where to use the Lies of P whistle isn't easy because the window you need to find isn't marked in obvious anyway. I had to basically walk around every wall and edge until I found what I needed - a single window where a child called Toma talks about playing with his friend, Murphy.
To save you the hassle of trying to work out what you need, here's everything I found on what to do with the whistle in Lies of P, and how to solve this odd little puzzle.
What to do with the Lies of P whistle
You'll find the faded whistle on a bench once you've defeated the Scrapped Watchman boss. It'll appear in your collectibles with a message about how it was blown by children while playing with a policeman puppet. You'll also get a Small Wooden Officer Puppet wooden toy. That comes with a lengthier description mentioning all four seasons which you can safely ignore because all you need to know is that the doll/police puppet was called Murphy.
Where to use the whistle in Lies of P
Once you have the whistle, head to a Stargazer and choose to teleport. You'll see the whistle marked against the 'Inside the House on Elysion Boulevard' fast travel point so head there. Go onto the balcony and turn right where you'll deal with a gun wielding puppet by a chimney sweep puppet ambush. Keep going and you'll be attacked by another chimney sweep turning a corner. Once you're on the roof look to the right for a ladder. The window you need for the faded whistle is visible below, brightly lit:
I found I had to walk right up to the window to get a prompt, there's literally no clue there's anything different about this particular window until you close enough. Unlike the weeping woman in Lies of P there's no silhouette or any indication there's anything to find here.
Get close enough though and you'll be able to speak to a child called Toma who'll talk about playing and missing Murphy. At this point go into your collectibles inventory and 'use' the whistle to blow it. Toma will be too ill to play but happy 'Murphy' dropped by and reward you with a Radiant Ergo fragment worth 500 Ergo, which can help you level up in Lies of P.
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