The 21-year-old was brought to the Seagulls on a permanent deal from Chelsea in the summer to work under Potter.
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However, it has not gone according to plan, with Chelsea acquiring the services of the former Ostersunds and Swansea City boss.
Despite backing the Scotland international to prove his worth, Clarke did see the funny side of the situation.
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"He's not very lucky Billy, is he?" Clarke said at a press conference.
"I think Billy was set in his own mind he wanted to play more football and he didn't feel he was going to get that chance at Chelsea.
"Football can change very quickly as Billy has found out. But he has got everything to play for at Brighton. They are going to appoint a new manager.
"It will be down to Billy. His performances in training, his performances in the matches will dictate whether he gets picked to play for Brighton.
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"It's down to Billy to do that. We've got him in the squad this time. We can give him a little bit of comfort when he comes up.
"I know Billy well. He's a good character. He's determined to make a big career in football and we know he is one for us.
"We will give him every support we can."
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It remains to be seen who is appointed the new manager of Brighton, but Gilmour will be keen to impress as he aims for regular minutes.
The midfielder has since been included in Clarke's Scotland squad for the upcoming Nations League matches against Ukraine and Republic of Ireland.