Jenny hopes that Mr Smith will suggest a good way to have her written English improved in a short

词句中谓语动词是哪个啊？
hope/ suggest / have 这些词在句子中分别做什么成分？

Jenny hopes that Mr Smith will suggest a good way to have her written English improved in a short period.
Jenny希望Smith先生会建议一个令她英文写作短期内提高的好办法。

主句很简单，是主谓宾结构：
Jenny(主) hopes(谓) that ...(宾语从句)

that引导的宾语从句又是主谓宾结构：
that Mr Smith(主) will suggest(谓，一般将来时) a good way(宾) to have ...(不定式作定语修饰a good way)

不定式：
to have (不定式非谓语动词) her written English(非谓语动词have的宾语) improved(过去分词非谓语动词作宾语补语补充说明her written English) in a short period(介词短语作improved的状语).

to have something done/improved/fixed/made/... 是一个固定用法。再给你举几个例子：

I'm going to have my hair cut.
John had all his money stolen.
How many authors have had their lives made into movies?