Cecil can remember the calls even now months into his relationship with Carlos (perfect hair, perfect everything Carlos). Sometimes, if he is alone late at the station and on a break, he will listen to the voicemail on his phone replaying those frantic messages left about clocks.
He can still remember his shock and excitement that bled into his words over the air when he realized the voicemails were from his crush. And now, even though Carlos has left other recordings about everything from groceries to saying he was working late, Cecil saves every single one.
In trying to tell himself it is /not/ at all weird to do this the radio host has fed himself plenty of reasons why he was keeping his lover's voicemails when he hears him every day. The first possible reason was that, like the man's hair, Carlos' voice was magic and beauty and just amazing. That is true of course but Cecil knew there was a deeper reason.
The next one he thought up was that, during the before-we-were-an-'us' period, those messages, about the clocks, was the only way he could hear his crush daily. And yes, that line of thought is also valid but again, not the concrete reason.
If he really, really thinks about it, Cecil keeps the messages (any of them) because it is the first time he can remember when someone reached out to /him/. Even if it began as a series of voicemails about clocks and their existence, Carlos' messages on Cecil's phone have become a daily reminder that he is important to more than just his listeners. He, Cecil Baldwin, is important to Carlos.
And really, that's all that matters in the end.