Shifting non-NA cells to the left

You can use the standard apply function:

df=data.frame(x=c("l","m",NA,NA,"p"),y=c(NA,"b","c",NA,NA),z=c("u",NA,"w","x","y"))
df2 = as.data.frame(t(apply(df,1, function(x) { return(c(x[!is.na(x)],x[is.na(x)]) )} )))
colnames(df2) = colnames(df)

> df
     x    y    z
1    l <NA>    u
2    m    b <NA>
3 <NA>    c    w
4 <NA> <NA>    x
5    p <NA>    y
> df2
  x    y    z
1 l    u <NA>
2 m    b <NA>
3 c    w <NA>
4 x <NA> <NA>
5 p    y <NA>

Thanks to @Richard Scriven for good observation

A) with is.na and order, lapply and rbind for aggregation

nosort.df<-do.call(rbind,lapply(1:nrow(df),function(x) { z=df[x,][order(is.na(df[x,]))];colnames(z)<-c("x","y","z");return(z) } ))

> nosort.df
  x    y    z
1 l    u <NA>
2 m    b <NA>
3 c    w <NA>
4 x <NA> <NA>
5 p    y <NA>

B) if sorted rows are required:

with sort, lapply and rbind

sort.df<-do.call(rbind,lapply(1:nrow(df),function(x) { z=sort(df[x,],na.last=TRUE);colnames(z)<-c("x","y","z");return(z) } ))

> sort.df
  x    y    z
1 l    u <NA>
2 b    m <NA>
3 c    w <NA>
4 x <NA> <NA>
5 p    y <NA> 

I have included a function for this task in my package dedupewider (available on CRAN). It allows to move NA to right, left or even top and bottom:

library(dedupewider)

df <- data.frame(x = c("l", "m", NA, NA, "p"),
                 y = c(NA, "b", "c", NA, NA),
                 z = c("u", NA, "w", "x", "y"))

na_move(df) # 'right' direction is by default

#>   x    y  z
#> 1 l    u NA
#> 2 m    b NA
#> 3 c    w NA
#> 4 x <NA> NA
#> 5 p    y NA

It implements the solution of reshaping data (from wide format to long and again to wide) and internally uses data.table functions. Thus it is significantly faster than standard solution using apply:

library(dedupewider)
library(microbenchmark)

df <- data.frame(x = c("l", "m", NA, NA, "p"),
                 y = c(NA, "b", "c", NA, NA),
                 z = c("u", NA, "w", "x", "y"))

df <- do.call(rbind, replicate(10000, df, simplify = FALSE))

apply_function <- function(df) {
  as.data.frame(t(apply(df, 1, function(x) c(x[!is.na(x)], x[is.na(x)]))))
}

microbenchmark(apply_function(df), na_move(df))

#> Unit: milliseconds
#>                expr      min       lq      mean    median       uq      max
#>  apply_function(df) 289.2032 361.0178 475.65281 425.79355 545.6405 999.4086
#>         na_move(df)  51.0419  58.1426  75.32407  65.01445  92.8706 216.6384