I have been studying the Bible and understand from the teaching of Jesus
that there is only one church. Is the correct? If so, should the
teaching/doctrine of the church be flawless because it is following the
Bible which is the flawless word of God?

You are absolutely right in saying that there is only one
church, biblically. One could debate whether Jesus taught this directly,
although Matthew 16:17,18 and John 17:20-23 certainly imply it. Perhaps
the key verse showing, biblically, that there is only one true church of
Jesus Christ is Ephesians 4:4, “There is one body…” In view of the
mind-boggling array of denominations as well as the great number of
avowedly non-denominational churches in the “Christian” world, it is
difficult to settle on exactly how to apply the statement that there is
one church. Perhaps this teaching can be summarized by stating that there
is only one church, and that this church is composed of all those who are
truly saved by the blood of Jesus Christ, wherever they may be. It seems
arrogant and unrealistic to say that any one religious group contains all
the saved people, or that in any one religious group, all those who would
claim membership are saved by Jesus.

All this does not really answer your question. Is there any
one group of disciples of Jesus who have absolutely flawless
teaching/doctrine? The answer is, obviously no! First of all, any group
which is striving to imitate Christ and to seek the truth with all their
heart will be constantly changing in response to what they are learning.
If, by some chance, this group happened to be perfect on some particular
date in time, they would cease to be perfect the next time they changed
their teaching on any matter, no matter how insignificant.

A better way to put it is to say that those who implement the
teaching in the Bible perfectly would have perfect teaching and doctrine.
Given that no one will ever reach that ideal;

1. There is not church with perfect teaching.

and

2. All who truly seek to serve God should be constantly in a process of
comparing what they believe and teach in order to strive to establish to
the best of their ability the true teachings of the Bible.

It is worth remembering that there are teachings which are
absolutely vital, such as who Jesus is and how one is saved by the blood
of Jesus. There are also teachings which are somewhat important, but
which are not a matter of salvation. We should give one another a break,
as I am sure God does as well. This is not an excuse for laziness and
lack of commitment to do what is right, but it is an acceptance of the
grace of God as a church. Even Paul accepted that not all would fully
accept his teaching, but encouraged those who were not there yet to live
up to what they already know is true. (Philippians 3:15,16). We should
strive to have the attitude of Paul as described in this passage.