The Demolished Path Lantern stands in the Valley of Ashes, a region that opens during main quest chapter 2, REGROUP. A Pistol is given in the toppled carriage about 70 m from the Lantern in a straight line, so pick it up before you move on. Other Lanterns of the region sit close by: The Climb is about 140 m away, Guidepost about 170 m and The Other Side about 220 m.
The chests nearest the Lantern hold an Indestructible Shell, Perfectionism and Superior Firearm Parts, and the closest Ambers give Indestructible Shell and Perfectionism. Around the Lantern you can also pick up an Epic Catalyst Clot, a Flaming Whetstone, a Refreshing Vial, Pistol Cartridges, Standard Firearm Parts and Crude Blade Material. Rest here first, then sweep the surrounding ground for these items.
Pistol
Indestructible Shell
Perfectionism
Superior Firearm Parts
Crude Blade Material
Epic Catalyst Clot
Flaming Whetstone
Pistol Cartridges
Refreshing Vial
Standard Firearm Parts
The Climb
Guidepost
The Other Side