When you play and save a round in Hole19 while Offline (not connected to the internet), that round will not sync automatically.
To see your list of saved rounds tap the 'Profile tab' (third at the bottom), then tap the 'Rounds Button' (Flag at center, under 'Following'). A round is not synced if you see a small yellow icon next to the score.
To sync a round, make sure you have a solid internet connection and do the following:
Open the unsynced round > Tap the '3 Dots Icon' in top-right > Select Sync Round.