He decided (and formed the intention) to carry on praying on his own because the sound of the imam’s voice was cut off, then the sound was restored, and he found that he had gone ahead of his imam in the prayer. What is the ruling?

Firstly:

If the sound of the imam’s voice is cut off during the
prayer, and the one who was praying behind him thinks that the sound will
not be restored, in this situation it is permissible for him to intend to
carry on praying on his own, without the imam, and to complete his prayer.

If that happens during Jumu ‘ah prayer, he should complete it
as Jumu‘ah (i.e., two rak‘ahs), if he has already prayed one rak‘ah with the
imam, and that loss of sound occurs in the second rak‘ah. But if it occurs
during the first rak‘ah, he should complete it as Zuhr (i.e., four rak‘ahs).

It says in Asna’l-Mataalib (1/225): 

If the wind closes the door (i.e. the door between him and
the imam) during the prayer, if he is able to open it immediately, then he
should open it and continue following the imam; otherwise, he should stop
doing so (and complete the prayer on his own). End quote.

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:

If the sound of the imam’s voice is cut off, and the
individual completes the prayer on his own, without the imam, then his
prayer is valid, because he is excused. But if we assume that this occurred
during Jumu‘ah prayer, and the sound was cut off during the first rak‘ah,
and the individual carried on praying on his own, without the imam, then he
should not pray Jumu‘ah per se, because he had not yet completed one rak‘ah
of it [and he should complete it as Zuhr, with four rak‘ahs, instead]. If
the sound was cut off in the second rak‘ah, and he carries on praying
without the imam, he should complete it as Jumu‘ah [with two rak’ahs],
because he had already prayed a complete rak‘ah of it. But if the voice of
the imam is cut off, the one who is praying behind an imam – whether the
individual is male or female – should not decide to carry on praying on his
own immediately; rather he should wait, because sometimes the sound is cut
off then they fix it. But if he is certain that it will not be restored,
then he should continue praying on his own.

End quote from al-Liqa’ ash-Shahri by Ibn ‘Uthaymeen.

It says in Fataawa al-Lajnah ad-Daa’imah (6/341):

If the sound of the imam’s voice is cut off and the
congregation can no longer hear it because of a malfunction in the sound
system, if the one who is praying behind the imam has already prayed one
rak‘ah of Jumu‘ah prayer with the imam, then the members of the congregation
should complete the prayer by themselves, each one praying on his own,
because it is no longer possible to follow the imam in the case of those who
cannot see the imam or those who are behind him. So if the sound of the
imam’s voice is cut off from those members of the congregation who cannot
see the imam or any of the worshippers who are behind him, before completion
of the first rak‘ah with the imam, then they should complete the prayer as
Zuhr (with four rak‘ahs)… End quote.

For more information, please see the answer to question no.
83009

Secondly:

If the sound of the imam’s voice is restored during the
prayer, and the one who was praying behind him had decided to continue the
prayer on his own, then he has the choice: if he wishes, he may resume
praying with his imam, and if he wishes he may complete his prayer on his
own.

If he resumes praying with his imam, and he had gone ahead of
his imam in the prayer, then if the imam is in the second rak‘ah of a four-rak‘ah
prayer, and the individual was in the third rak‘ah – for example – then when
the imam reaches the third rak‘ah, which is the fourth rak‘ah for that
individual, he should sit down and recite the tashahhud, and not get up with
his imam (for the last rak‘ah). Then he has the choice in this case: if he
wishes, he may say the tasleem and stop following the imam, or if he wishes
he may wait for his imam and say the tasleem with him.

An-Nawawi (may Allah have mercy on him) said in al-Majmoo‘
(4/209):

If the prayer of the individual is completed first, then it
is not permissible for him to follow his imam in any additional rak‘ah;
rather, if he wishes he may stop following the imam once he has completed
the prayer, and he may recite the tashahhud and tasleem, and his prayer will
be valid, with no difference of scholarly opinion, because he stopped
following the imam for a reason that had to do with the prayer; or if he
wishes he may wait during the tashahhud, and offer a lengthy du‘aa’, so that
the imam may catch up with him, then he may say the tasleem after the imam.
End quote.

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said in
ash-Sharh al-Mumti‘ (2/314):

Question: if the one who was praying behind an imam begins to
pray on his own due to a valid excuse, then that excuse no longer applies,
can he go back to praying with the imam, or should he continue praying on
his own?

The fuqaha’ said: It is permissible for him to go back to
praying with the imam, or to continue praying on his own.

If we assume that he prayed on his own and prayed one rak‘ah,
then went back to praying with the imam, but the imam was still in the
rak‘ah in which the individual decided to pray in his own, so the imam is
one rak‘ah behind him, then when the imam gets up to complete his prayer, he
may remain sitting and wait for him, or he may continue praying on his own
and complete his prayer. End quote.

And Allah knows best.