Below is a fix (at least for me) for sending email on my GoDaddy account:
With over 48,000 views to this page, obviously soooooo many people are having a problem with GoDaddy's outgoing server. And trust me, it's GoDaddy not Apple that is the problem. The GoDaddy outgoing server makes your "sent mail" work. If you're having "sent email problems" it's because the Godaddy outgoing server isn't linking up with your email in the Mail program.
I finally had a GoDaddy help person admit to me that they have been having outgoing server problems for years, especially with their "POP" email accounts. POP is the oldest of the 3 ways to send mail (POP, IMAP, and Exchange). The rep told me that they (GoDaddy) will probably end offering POP accounts next year because it's such an "outdated" method of sending email. He suggested that I switch to an "Exchange" way of sending email (at an additional cost), which is the newest and most reliable. That was after he couldn't come up with any way of fixing my POP email. I told him no way, that I would play around until I fixed it myself.
And I finally did... after trying everything, and I mean everything (I'm a web designer). This is how I finally found a way for my POP email account to work and send email:
1 - Sign in to your GoDaddy account
2 - In the "Email" section, click the green rectangle "Launch" button
3 - A new page will open showing your email(s)
4 - The email line will have 6 columns: Email Address, Attributes, Alerts, Plan Name, Usage, Relays
5 - Look at the last column titled "Relays" and see if it has anything it in;
in most cases it should say "0/250" but it also may have nothing in it
6 - Relays are the number of emails you are allowed to send out each day on that particular email account
7 - Regardless if you have "0/250" or nothing in the "Relay" column,
click on the name of the email, in the "Email Address" column, and a new window will open titled "Edit Account"
10 - Go down near the bottom, to the "SMTP relays per day" line
11 - Even if it shows 250, it still may not be showing up in the 6 column line, in the "Relay" column
12 - So... instead of the 250 number, use the disclosure triangle and choose 0, and then hit the black "Save" button
13 - Then go back again, and do the same thing, but this time choose 250, and hit "Save"
14 - Then see if the 0/250 shows up in the 6 column line, in the "Relay" column
15 - You may have to do this several times, but keep trying, until the "0/250" finally does show up in the "Relay" column
16 - When it does, open your Apple Mail program, and try to send yourself an email
Hopefully, your mail will now work. It did for me when I did all this. Good luck!