There are three usual reasons why this may be happening:
Your Covve account is already in sync with another address book such as gmail or outlook. Each Covve account is synchronized with a signle address book. If you'd like to sync with a different address book you can create a new account.
Tip: A nice way to synchronise Covve with both your iPhone and other address book (e.g. gmail or outlook) is to add these other address books as contact sources to your iphone and then synchronise Covve with your iphone.
When you signup to Covve from your iPhone you choose which of the contact groups/sources from your iPhone you want Covve to synchronize with. If you don't choose all your sources then some contacts from the iPhone address book won't sync with the Covve app.
To see which sources/groups you have on your iPhone go to your iPhone contacts and click on groups at the top left:
If instead of "Groups" you see a refresh icon that means you only have a single group.
For more information on how to manage groups and how to change them on Covve see here.
When you add a contact in your iPhone's address book it should appear in your Covve app.
If this isn't happening it is likely that you've set your iPhone to save new contacts to a certain group which you're not synchronizing with the Covve app.
You can change which group your new iPhone contacts are being added from your iPhone's "Settings", then "Main, Contacts, Calendars" and then changing "Default Account"