"Yet, due to the interference of the 'Clown,' the Cross Guild embarks on an unexpected journey... What does 'Hawk Eyes' seek? Peace, or perhaps the 'One Piece'?"
"He dismisses idle chatter with a laugh but agrees with hiding in the shadow of a Yonko. However, Buggy's declaration throws Mihawk's strategy into disarray."
Nothing but sheer illiteracy.
There is not a single shred of evidence anywhere except the voices in your head... which says Mihawk dreamt of being a PK and then failed.
He is being forced against his will (just like Crocodile) to participate in the current race for One Piece due to Buggy's interference.
We know that because of buggy cross guild wants to go after the one piece
What does 'Hawk Eyes' seek? Peace, or perhaps the 'One Piece'?"
This sentence specifically is asking what mihawk himself specifically wants
Even ignoring the vibre card the manga itself heavily implies that mihawk was also after it back in his youth
Buggy’s speech being directed at mihawk and cross guild as a whole. With him being in a crew full of people who gave up on their dream of achieving the one piece from buggy to crocodile and even all the fodder
Buggy shits on crocodile and mihawks utopia plan calling it small time
Buggy asking them what they wanted to be back when they were young heavily insinuating they also wanted to become pirate king
Buggy’s speech obviously catching their interest and attention
Buggy’s speech was obviously directed at them including the rest of cross guild
On top of that mihawk being at Roger’s execution on a spread full of pirates that were also after the one piece