Please help me to analyze Charlotte Turner Smith's sonnet "Written in October."

Knowing that the poem is written by Charlotte Turner Smith is actually a big help in understanding this sonnet.  Smith pioneered a new sonnet form called the "elegiac sonnet."  This sonnet form is also a bit of an amalgamation of previous sonnet forms in that Smith uses the rhyme scheme made famous by Petrarch; however, she doesn't always stick to it exactly.  She will occasionally blend in Shakespeare's rhyme scheme or even make use of "eye rhymes."   
"Written in October" is an elegiac sonnet.  It is a fourteen-line poem written in iambic pentameter.  That means each line contains five separate iambic feet.  An iambic foot is a poetic unit made up of an unstressed syllable followed by a stressed syllable.  The word "allow" is a word with this kind of rhythm.  Line 5 is a good example of iambic pentameter.

Are well / attuned / to my / dejec / ted mood....

Smith's rhythm and meter are not perfect, though.  Remember, this sonnet form is intentionally mixing things up.  Line three is not written in pentameter.  It has too many syllables.  

And muttering many a sad and solemn sound....

"Written in October" also has a rhyme scheme that is different from Petrarch and Shakespeare. Petrarch's rhyme scheme is ABBAABBA CDECDE.  It's typically divided into two stanzas, too.  The first stanza is eight lines (an octet), and the second stanza is six lines (a sestet). Shakespeare's rhyme scheme is ABAB CDCD EFEF GG and is divided into three quatrains that end with a couplet. Smith's poem doesn't follow either form.  It begins with an ABAB rhyme scheme, but then it moves to a CDDC rhyme scheme.  Lines 9–12 are EFFE, and the poem does end with a couplet.  
As with the meter not being perfect, the rhyme scheme isn't perfect in this poem either.  It only looks that way upon first glance.  Lines 5 and 8 are "eye rhyme" lines.  Eye rhymes are words that look like they should rhyme because of spelling, but they are pronounced differently.  Line 5 ends with "mood" and line 8 ends with "wood."  They differ in spelling by only a single letter, but they are pronounced quite differently.  The final couplet is an eye rhyme as well. Smith definitely messes with reader expectations about sonnets just by diverting from traditional forms of rhyme and meter.
This sonnet is about the narrator's dejected mood. The first four lines tell readers about the depressing parts of the fall season.  Leaves are dead, fallen, and faded, and other things are pale and dried out.

The blasts of Autumn as they scatter round


The faded foliage of another year,


And muttering many a sad and solemn sound,


Drive the pale fragments o'er the stubble sere....

Line 5 then tells readers that the dying image of fall is perfect for the narrator's mood. Line 6 further stresses this point by saying that the fall mood is by far better than the hopeful outlook of spring.

(Ah! better far than airs that breathe of Spring!)

In lines 7–12, the narrator sets up a comparison.  The narrator says that he/she would rather listen to hoarse-sounding birds seeking shelter in leafless forests than listen to sounds that are typical of May.  
The final couplet is quite depressing.  

Nature delights me most when most she mourns,
For never more to me the Spring of Hope returns!

The lines echo the poem’s beginning by reminding the reader that the sorrowful effects of fall are much more appealing to the speaker than other seasons—an unconventional view that also defies reader expectations.  It’s the poem’s final line that is extremely depressing.  The narrator admits that hopeful feelings will never again return to him/her.