Meaning of to "will someone on"

Solution 1:

Though one answer analyses this as an example of the single-word verb 'will', and this probably explains the development of the construction, Wikipedia considers 'will on' a transitive multi-word verb (it uses the term 'phrasal verb').

Wiktionary has a definition for the term:

will on Verb

[Categories: English lemmas ... English verbs ... English phrasal verbs ... English phrasal verbs with particle (on)]

to will on

To wish intensely that someone succeeds in what they are doing. Often implies a silent, or almost inaudible wish.

[reformatted]

There is often debate over whether some of these expressions should be considered as unitary [multi-word verbs] or not.

Solution 2:

In this context, to will means something like

[WITH OBJECT AND INFINITIVE]

Make or try to make (someone) do something or (something) happen by the exercise of mental powers:

'reluctantly he willed himself to turn and go back'

'she stared into the fog, willing it to clear'

Your example doesn't include the infinitive, but rather the small clause construction "you on", which can be read as something like "you to go on".

Thus, Cameron is saying something like:

But I will be intending/wishing/urging you to go on.

Solution 3:

But I will be willing you on.

This means "I will be wishing for you to progress/advance."

"Willing something/someone on" is like when your team has the ball and is advancing and you try to "wish" them forward to a goal. Sometimes it implies physically cheering them onward, sometimes the "cheering" is unspoken and the most an observer of you might see is teeth gritting and a stiffening of the posture.

Solution 4:

To will something is to [try to] cause or change by an act of will or volition. Merriam-Webster (third usage)

will - transitive verb - decree, ordain

This is something you can do, whether or not you have to authority or power to follow through: I can will my bowling ball to turn back toward the center of the lane, even as it tips into the gutter.

It indicates what you hope to happen, so Mr. Cameron will be hoping the Tories do well and continue on. Whether he exercises any force other than the force of his own desire and verbal support is yet to be known.