outboard sputters on way out, and then is fine all of a sudden

The last two trips we took, our motor would start off OK, and then start to lose power. The first time it didn't do it until we had been riding for about 15-20 minutes. We had it full throttle and were only going 9 mph (usually goes close to 40). SO we stopped to fish (since we were already out). Started it up to put-put back in, and it was fine so we drove in full speed.

We thought it might have been a bad fuel line so we replaced it. Went out this past weekend and it did the same thing, but this time it started losing power just about a minute into it. We didn't have very far to go to our fishing spot so we still went. When we were done we started motor up and was able to come in full speed.

I am going to bring it to an outboard mechanic this week, but wanted to see if anybody on here has any idea of what might be happening based on what you read. TIA