Hey there miamitopa,
Welcome to the Apple Support Communities! From what it sounds like, you're running into some issues with playing purchased content. Maybe I can give you a bit of guidance that will help.
On which device does this issue occur? For example, is it a Mac, a PC, or some other device that uses iTunes? Do you have a different device that you can attempt to listen to those same songs to see if the same issue arises? This could be another computer or an iPhone, iPad, or iPod touch.
When you do enter your account information, are you able to play those songs? Have you checked to ensure that the Apple ID signed into iTunes is the one that was used to purchase those songs? If not, that'd explain why you need to enter the credentials each time. You can check the signed in account by clicking the Account menu in iTunes. If it differs from the one used to purchase those songs, choose Sign Out from under the Account menu and then sign in with the correct Apple ID.
If you need to check which account made that purchase, you can See your purchase history for the App Store or iTunes Store.
Cheers!