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It is a wonderful topic that will follow up to take advantage of it

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How Africans Are Saving Their Own Soil

By April Fulton

An archaeologist stumbles on a hidden farming practice that's been passed down through generations. Here's how it works.

 For hundreds of years, parts of sub-Saharan Africa have suffered from poor soil. Weather, shifting populations, and slash-and-burn practices have left wide swaths of land relatively useless for growing food without major commercial intervention. But that’s not the whole story of African farming.

 Fairhead, who has been exploring settlements in the forests of West Africa since the 1990s, had for years observed locals planting crops on the grounds of former villages. As an archaeologist digging for historic artifacts in the same locations, it could be something of a nuisance, he acknowledges, but he started to wonder why it was happening.

He looked for scientific literature on African soils and turned up nothing until he stumbled upon a similar discovery of soil improvement in the Amazon as far back as 5,000 years ago.

 .....

If residues of firewood and food accumulate so too do other elements, including P

I agree with the statement given by Dr. Paul. We may only guess it as there was no literature available or no estimation procedure at that time.

Your question focuses on two points: "Paleolithic" and "Open air sites". If you are interested only in open air Paleolithic sites, these are quite fragile to the passage of time being readily exposed to the elements and the effects of post prepositional processes that directly affect phosphate concentrations of any kind. However there are examples that directly or indirectly involve the analyses of phosphates in Paleolithic cave sites. Cave sites are not "open air" obviously, but constitute close systems that involve specific sedimentary, taphonomic and diagenetic processes that affect phosphates concetrations and their preservation differently. Phosphate "signatures" are still distinguishable in caves because of these specific processes. Such research for instance, has been conducted in Paleolithic cave sites assessing different functional areas or degree of preservation etc.. :

Schiegl et al (2003) Paleolithic Burnt Bone Horizons from the Swabian Jura: Distinguishing between In Situ Fireplaces and Dumping Areas. Geoarchaeology: An International Journal, Vol. 18, No. 5, 541–565 (2003)

Karkanas et al (2007) Evidence for habitual use of fire at the end of the Lower Paleolithic: Site-formation processes at Qesem Cave, Israel. Journal of Human Evolution 53 197e212

Shahack-Gros et al (2004) Bat guano and preservation of archaeological remains in cave sites. Journal of Archaeological Science 31 (2004) 1259e1272

This may not be exactly what you were looking for, yet i hope it was useful to you somehow.

Well, thank you Mr. Roussos. Especially the Schiegl paper is very helpful. The thing is, that when it comes to cave sites, the phosphates are mainly down to bats, as also suggested by the third paper you sent me. Will be interesting to see, what happens at open-air sites. Anyway, thanks again for your reply.