Clarke's men travel to QPR for a third round tie on Saturday and, while their hosts remain at the bottom of the Premier League table, West Brom are an impressive seventh, despite their New Year's Day loss to Fulham.
Clarke said on Friday: "The FA Cup is a competition we want to try and do well in. I've enjoyed success as a player and a coach in the FA Cup and I'd like to continue in the same vein.
"We are only involved in three competitions and we went out of the League Cup early but we'd like to stay in the FA Cup as long as possible.
"I assume QPR's priority will be staying in the Premier League and they had a great result at Chelsea (a 1-0 win on Wednesday) which means they will be a bit more buoyant tomorrow.
"We are a little more comfortable in the league and are sitting in a reasonable position, one we want to try and improve on in the second half of the season, but the FA Cup is a competition we want to do well in."
Some West Brom supporters booed the side at half-time and full-time of the 2-1 home defeat to Fulham, a reaction which Clarke felt was harsh.
"I was disappointed with the crowd's reaction," he said. "During the first 45 minutes, we were below the standards we have set at home this season.
"I could understand the boos at half-time but I couldn't understand the boos at the end because, in the second half, we gave it a right good go.
"I don't think there is any reason for the crowd to have a go at the team after the way we have played, especially in home games this year.
"The players go out every week and try and give their best but some weeks it doesn't happen and that is the nature of football. You don't have a divine right to be better than the opposition."