你好这位朋友,感谢你的信任,你的求助我已经收到了.解答如下：
increasingly --> increasing：正确.increasing本身就是形容词,用来修饰number,加上ly是不行的,因为副词不能修饰名词
As引导的是从句,在第一个逗号之前,都是这个从句的领地.在这个领地内,需要属于从句自己的动词.修改如下,请看大写：As an increasing number of people ARE moving into
city前需要加定冠词THE,jobs已经有了s是复数泛指所有的工作岗位,不需要定冠词.city之所以需要THE是因为需要特指那些找工作的人的新目的地.
big cities are后面有三种说法行得通,你可以选其一：
1 - big cities are FACED WITH large pressure
2 - big cities ARE FACING large pressure
3 - big cities FACE large pressure
第二句开头是固定用法：On THE other hand,这里的定冠词不可省略,特指事情的另一面,表示转折.
逗号后面语意紊乱,需要稍作修改.接下来我把两句话统一帮你修改和改进之后如下,请注意比对：
As an increasing number of people are moving into the city for jobs, big cities are facing GREATER* pressure. On the other hand, companies are offering inadequte positions to meet these job seekers' needs in middle-small sized cities.
*处注这里我用了greater这个比较级,和之前的increasing更加呼应.
希望对你有帮助:-)