give weight to something
add weight to something

City officials said they plan to give more weight to experience next time.

Did the agency give too much weight to the few?

Nor, they said, does it give enough weight to the financial needs of second families.

Each of these things seems to be given equal weight.

Even under the new rules, however, there was still room to give added weight to experience.

"Why isn't this very significant evidence, to which we must give great weight?"

These emotions in a 5-year old were not given too much weight then.

Human rights, on the other hand, should not be given too much weight, the immigrants said.

No reasonable man would give much weight to those, in any case.

It gives due weight to action research as a process rather than just a set of things to do.