The gemara seems to understand that one can spread out an aveida for it's need but not for personal needs. But, the gemara questions whether one can spread out an aveida when it is לצרכו ולצורכה, both for the aveida need and for his personal need. The gemara tries to prove that it is permitted, but is unsuccessful in being able to prove it.
The Mishna seems to weigh in on this issue and clearly holds that לצורכו ולצורכה is permitted, because the mishna allows one not only to air out a sefer but even to read from it. Clearly, the mishna is permitting even personal use, so long as it has to be done for the purpose of the lost object. Tosafos explains that the gemara is reluctant to permit לצורכו ולצורכה because one is likely to do it for ONLY personal needs which would surely be assur. The prohibition of לצורכו ולצורכה is not me'ikar hadin, rather it is a d'rabonon gezeira that one shouldn't use it לצורכו ולצורכה because it may lead to using it for personal use. Therefore, by a sefer where the mishna gives specific instructions how and when to use it, there is no concern that he will take advantage so the rabbonon never imposed an issur. Based on this approach, the gemara that tries to prove that it is assur from the case of Eglah Arufa is difficult to understand. How does the fact that egla arufa becomes passul m'doraysa לצורכו ולצורכה, prove that the rabbonon are concerned that if one uses an aveida לצורכו ולצורכה it will lead to using it for only personal use? Tosafos will be forced to learn the question that the prohibition of לצורכו ולצורכה is that the rabbonon were afraid that you may come to use it only for personal use and therefore imposed a p'sul even when it is done לצורכו ולצורכה as if it is entirely being used for personal use. Here too, we should make a gezeria that even לצורכו ולצורכה is considered as if it were only used for personal use and pasul. The gemara answers that the p'sul is d'oraysa and therefore has no connection to our question.
The chofetz chaim (ahavas chessed 7 - footnote 2) understands that the Mordechai and rashi in pesachim disagree with Tosafos and hold that לצורכו ולצורכה is assur m'ikar ha'din, not just a gezeira that it will lead to personal use. Meaning, we view לצורכו ולצורכה on a d'oraysa level as if it were being used only for personal use and therefore assur. This is a tremendous chiddush because the action that is being done is entirely permitted, just that the intention is assur and would render the entire action an issur. But, based on this approach the proof from egla arufa works out better.