Clarence W. Murphy certainly was a prolific songwriter, and in a short period of time, too. I've compiled a list of every song written by him that I can find a mention of on the Internet. Many of these came from one site:
http://www.truesoundtransfers.de/edi412.htm
To keep the list simple, I have always listed his name as "Clarence W." although I sometimes found it as "C.W."; and I always listed his name first although it was sometimes given second. Probably the most frequently mentioned songs are "Has Anybody Here Seen Kelly?" and "Let's All Go Down the Strand".

CLARENCE W. MURPHY:
Boss of the House – 1910
Goo-Goo Land – 1909
Hogmanay – 1912
I Live in Trafalgar Square - 1898
In Your Old Tam-O'-Shanter – 1910
O'Brien – 1911
The Tablet of Fame - ?
We Must Have a Song about the Isle of Man – 1912
You Must Come Round on Saturday – 1912

CLARENCE W. MURPHY & ? ALLEN:
Let's All Go into the Ballroom - 1912

CLARENCE W. MURPHY & CHARLES COLLINS:
Come and Hear Him Play His Oom-Tera-Ra – 1909
On the Same Place Every Time - 1910

CLARENCE W. MURPHY & DAN LIPTON:
Beautiful, Beautiful Bed - 1915
I Only Got Married To-Day - 1909
I Shall Sulk – 1910
I'll Be Cross, Arabella - 1909
I've Got a Garden in Sweden - ?
No Wonder I Look Jolly – 1911
Oh, Oh, Antonio - ?
Plink Plonk, Or: The Skin of a Spanish Onion - 1909
Shirts – 1910
The First Time That I've Been in Love - 1910
The Ugly Duckling - 1904
We Don't Want More Daylight - 1909
We'll Treat You like One of Our Own – 1910
You Must All Do as I Do – 1912

CLARENCE W. MURPHY, DAN LIPTON & JOHN NEAT:
Lassie from Lancashire - ?

CLARENCE W. MURPHY & GEORGE ARTHURS:
Put Me amongst the Girls - 1907

CLARENCE W. MURPHY & HARRY CASTLING
Have a Banana!* – 1910
I'm the Man That Buried Flanagan – 1911
Kissing Time - ?
Let's All Go Down the Strand* - 1904
Meet Me, Jenny, When the Sun Goes Down - 1908
Pull Yourselves Together, Girls - 1909
The Girl in the Clogs and Shawl - 1909
The Singer Was Irish – 1910
We All Go the Same Way Home – 1911

CLARENCE W. MURPHY & W HARGREAVES:
Little Yellow Bird - ?

CLARENCE W. MURPHY & WILL LETTERS:
Flanagan – 1910
Has Anybody Here Seen Kelly? – 1904
Hi, Hi, Hi, Mr. McKie - 1911

CLARENCE W. MURPHY & WORTON DAVID:
Follow The Footsteps In The Snow - 1912
Have You Heard John James O'Hara? – 1911
Hold Your Hand Out You Naughty Boy – 1914
Keep Your Head Down, Fritzi Boy (We Saw You) - 1918
They're All Single by the Seaside – 1911

CLARENCE W. MURPHY, NORA BAYES, AND JACK NORWORTH
How Can They Tell that I'm Irish? - ?

*Let's All Go Down the Strand---Have A Banana!
Harry Castling (d. 1930) "From the song, 'Let's All Go Down The Strand' (1904), written with C. W. Murphy. The words 'have a banana' were interpolated by audiences. Although not part of the original lyrics, they were included in later versions." (From Rees Cassell, "Companion to Quotations" (1997))