1. to pick up the pieces, to sort things out, cleaning up the mess Iconspeaker_3
[puin-rui-men, ruim-de puin, puin-ge-ruimd]Puinhopen

The literal translation of "puinruimen" is covered in 2. below. So as long as you are not talking to a construction worker, you can stick to this translation 🙂

Note that "puinruimen" usually implies the negative side of "sorting things out".

– "Hoe was je eerste dag na de vakantie?" – "Ach je weet wel, e-mail lezen en puinruimen." 
("How was your first day after the holidays?" – "Oh you know, reading e-mails, picking up the pieces…")

– "Na een relatie van 15 jaar waarin zij zichzelf volledig heeft weggecijferd, is het nu een kwestie van puinruimen." 
("After a relationship of 15 years in which she completely effaced herself, it is now a matter of picking up the pieces.")

– "Ik heb het project overgenomen van Frank. Nou, ik denk dat ik de eerste maand voornamelijk aan het puinruimen ben!" 
("I took over a project from Frank. My, I think that the first month I’ll be mostly cleaning up the mess!")

Related words:
– Puin: debris, rubble, mess [noun] [het puin, <no plural>].
– Ruimen: to clear out [verb] [ruimde, geruimd].
– Puinhoop: pile/heap of rubble/debris [noun] [de hoop, de hopen]. This is also used figuratively.
– Puinzooi: a real mess [noun] [de puinzooi, <no plural>]. Note that "zooi" already translates to "mess/rubbish" so "puinzooi" is really bad 🙂

2. clear up the debris/rubble Iconspeaker_3
[puin-rui-men, ruim-de puin, puin-ge-ruimd]Puinruimen

This is what construction workers do, or whoever you find willing.

– "Je kunt daar niet met de auto langs; ze zijn aan het puinruimen." 
("You can’t pass there with your car; they are clearing debris.")

– "Ruimen ze puin, of zijn ze aan het puinruimen?" – "Huh, is dat niet hetzelfde dan?" 
("Do they clear debris, or are they clearing debris? – "Huh, isn’t that the same?" There is a difference similar to the difference in English. The "aan + het + infinitive" is typically used for an activity that continues for some time. )