The phrase "give higher priority to" is correct and usable in written English.
It is typically used to describe something that is given more importance or attention than other things. For example: "We must give higher priority to the safety and health of our employees."
It could, for example, tap into social networks to give higher priority to messages coming from senders with close ties to their recipients.
They also help explain why, according to McElhaney's research (pdf), companies with higher female representation on their boards tend to give higher priority to environmental and social issues.
As a consequence, the UK is heading down the gender equality league table, as other countries do more and give higher priority to reversing their own inequalities.
The results (see table 3) show that in the United States and a few other countries academics still appear to give higher priority to teaching than to research.
City officials will also give higher priority to training to try to prevent friendly fire incidents and will likely bring in outside experts to assess the effectiveness of the department's programs.
Verizon, Comcast, AT&T and the other carriers will inevitably give higher priority to traffic from companies that pay – a double-dipping process for the carriers – and lower priority, if any at all, to the ones that can't or won't pay.
Academies and free schools, like all schools, must give highest priority to looked-after children.
"It would not surprise me if we give high priority to youth homeless programs".
I was encouraged by my colleagues in the Ministry of Justice to urge him to give high priority to the governance agenda.
In this connection, the sides give high priority to Russia's accession to the World Trade Organization on standard terms.
The guidance says schools should "give highest priority to the consideration of cost and quality for parents".
