It means '特意'  or '特地' in Chinese.

with the special intention of

Here is a word for word translation:

特+意:  special + intention

专门看=特意看=特地看

特地 [tè dì]

• • 1.
    for a special purpose; specially; go out of one's way:

    come specially to ask for advice;特地来请教

    I came here specially to see you.我特地来看你。

专门 + 看 （或者别 动词）

专门 literally means "specialized door".  In the old days, entrances to a household (the wealthy and nobility) have special functions.  The grand entrance is reserved for special occasions or receiving (very important) guests.  The side door is used for other daily purposes.  Hence, depending on your purpose, you use the appropriate door.  Of course, if you are going an important household, you have an important purpose.  Hence, today it means, on a specific mission or cater to to some specificity invovling some sort of travel.

The use of the term is not appropriate in your example as there is no implication of "movement".  At the same time, it is used in a non-negated way.  For example, 我专门不去....  because of the 不.  For this purpose the word 特意 is used instead, 我特意不去....  Quirks of the language.

As an adjective, it means "specialty" as in 专门店, "specialty store".