The verb पाना can be used to mean “to get”, “to obtain”, “to achieve”, “to attain”, “to find”, etc.
मैंने पाया है कि हिंदी सीखना मुश्किल तो है पर असंभव नहीं – “I have found that it’s difficult to learn Hindi, but not impossible”
मैंने अपनी किताब फ़र्श पर पड़ी हुई पाई – “I found my book lying on the floor”
मैंने उनसे सम्मान पाया – “I earned their respect”
However, this verb is most commonly used as an auxiliary verb. When used as an auxiliary, it is similar to सकना. It means “to be able”, “to manage to”, etc.
शोर के वजह से मैं सो ही नहीं पाया – “Because of the noise, I couldn’t even sleep”
मैं अपना काम ख़त्म नहीं कर पाऊंगा – “I won’t be able to finish my work”
मैं खाना नहीं खा पाया – “I couldn’t manage to eat the food”
Most transitive, perfect / perfective verbs exhibit ergative alignment. In other words, in the example “मैंने अपनी किताब फ़र्श पर पड़ी हुई पाई”, the agent, मैं, is followed by ने, and the verb, पाई, agrees with the object, किताब (which is feminine and singular). This is the usual pattern, and so it is not surprising.
When compound verbs are used, the general principle is that if both verbs require ने, then the compound verb require ने also. However, पाना is a notable exception. When it is used as an auxiliary verb, the compound verb does not require ने. Thus, in the example “मैं खाना नहीं खा पाया”, ने was not used. If the speaker were female, then the sentence would be “मैं खाना नहीं खा पाई”.