What did the Pan-African Conference hope to achieve by sending this petition to the Queen of England?

I assume that you are asking about the petition that was
sent to Queen Victoria of England by the delegates of the Pan-African Conference that
was held in London in 1900.  The purpose of this petition was to try to get colonial
powers in Africa to treat their African subjects more
humanely.

The petition asked the Queen investigate and to
work to ameliorate the plight of Africans in African colonies.  It asked her to look
into laws and customs that discriminated against them.  It also asked her to investigate
things like the working conditions that they faced in the
colonies.

Overall, then, the goals of the petition was to
get the colonial powers to be more humane and to bring the injustices that Africans
faced to the attention of the general publics of the colonial
powers.